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Your data matches 13 different statistics following compositions of up to 3 maps.
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Matching statistic: St000999
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000999: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000999: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1,0,1,0]
=> 1
([],2)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(0,1)],2)
=> [2]
=> [1,1,0,0,1,0]
=> 1
([],3)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
([(0,2),(1,2)],3)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([],4)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
([(1,3),(2,3)],4)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([],5)
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 1
([(3,5),(4,5)],6)
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 1
Description
Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001009
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001009: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001009: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
([],2)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(0,1)],2)
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
([],3)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
([(0,2),(1,2)],3)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
([],4)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
([(1,3),(2,3)],4)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([],5)
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 1
([(3,5),(4,5)],6)
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 1
Description
Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000771
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([],4)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
([],5)
=> ([],5)
=> ([],1)
=> ([],1)
=> 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 1
([(4,5)],6)
=> ([],5)
=> ([],1)
=> ([],1)
=> 1
([(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(3,6),(4,5)],7)
=> ([],5)
=> ([],1)
=> ([],1)
=> 1
([(2,3),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,6),(2,5),(3,4)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000772
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([],4)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
([],5)
=> ([],5)
=> ([],1)
=> ([],1)
=> 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 1
([(4,5)],6)
=> ([],5)
=> ([],1)
=> ([],1)
=> 1
([(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(3,6),(4,5)],7)
=> ([],5)
=> ([],1)
=> ([],1)
=> 1
([(2,3),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,6),(2,5),(3,4)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St000777
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([],4)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
([],5)
=> ([],5)
=> ([],1)
=> ([],1)
=> 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 1
([(4,5)],6)
=> ([],5)
=> ([],1)
=> ([],1)
=> 1
([(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(3,6),(4,5)],7)
=> ([],5)
=> ([],1)
=> ([],1)
=> 1
([(2,3),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,6),(2,5),(3,4)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St001645
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([],4)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
([],5)
=> ([],5)
=> ([],1)
=> ([],1)
=> 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? = 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 1
([(4,5)],6)
=> ([],5)
=> ([],1)
=> ([],1)
=> 1
([(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1
([(3,6),(4,5)],7)
=> ([],5)
=> ([],1)
=> ([],1)
=> 1
([(2,3),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(1,6),(2,5),(3,4)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 1
Description
The pebbling number of a connected graph.
Matching statistic: St000259
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([],2)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([],3)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([],4)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1 - 1
([],5)
=> ([],5)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 3 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 1 - 1
([(4,5)],6)
=> ([],5)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 3 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(3,6),(4,5)],7)
=> ([],5)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,3),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,6),(2,5),(3,4)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000260
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([],2)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([],3)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([],4)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1 - 1
([],5)
=> ([],5)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 3 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 1 - 1
([(4,5)],6)
=> ([],5)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 3 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(3,6),(4,5)],7)
=> ([],5)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,3),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,6),(2,5),(3,4)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000302
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([],2)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([],3)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([],4)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1 - 1
([],5)
=> ([],5)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 3 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 1 - 1
([(4,5)],6)
=> ([],5)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 3 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(3,6),(4,5)],7)
=> ([],5)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,3),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,6),(2,5),(3,4)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
Description
The determinant of the distance matrix of a connected graph.
Matching statistic: St000466
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([],2)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([],3)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([],4)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1 - 1
([],5)
=> ([],5)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 3 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? = 1 - 1
([(4,5)],6)
=> ([],5)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 3 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 2 - 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? = 1 - 1
([(3,6),(4,5)],7)
=> ([],5)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,3),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,6),(2,5),(3,4)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
Description
The Gutman (or modified Schultz) index of a connected graph.
This is
$$\sum_{\{u,v\}\subseteq V} d(u)d(v)d(u,v)$$
where $d(u)$ is the degree of vertex $u$ and $d(u,v)$ is the distance between vertices $u$ and $v$.
For trees on $n$ vertices, the modified Schultz index is related to the Wiener index via $S^\ast(T)=4W(T)-(n-1)(2n-1)$ [1].
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