Your data matches 8 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000288
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => 1 => 1 => 1
([],2)
 => [2] => 10 => 11 => 2
([(0,1)],2)
 => [1,1] => 11 => 11 => 2
([],3)
 => [3] => 100 => 101 => 2
([(1,2)],3)
 => [1,2] => 110 => 111 => 3
([(0,2),(1,2)],3)
 => [1,1,1] => 111 => 111 => 3
([(0,1),(0,2),(1,2)],3)
 => [2,1] => 101 => 110 => 2
([],4)
 => [4] => 1000 => 1001 => 2
([(2,3)],4)
 => [1,3] => 1100 => 1101 => 3
([(1,3),(2,3)],4)
 => [1,1,2] => 1110 => 1111 => 4
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => 1101 => 1110 => 3
([(0,3),(1,2)],4)
 => [2,2] => 1010 => 1101 => 3
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => 1111 => 1111 => 4
([(1,2),(1,3),(2,3)],4)
 => [2,2] => 1010 => 1101 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => 1111 => 1111 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => 1101 => 1110 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 1011 => 1101 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => 1001 => 1010 => 2
([],5)
 => [5] => 10000 => 10001 => 2
([(3,4)],5)
 => [1,4] => 11000 => 11001 => 3
([(2,4),(3,4)],5)
 => [1,1,3] => 11100 => 11101 => 4
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => 11010 => 11101 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => 11001 => 11010 => 3
([(1,4),(2,3)],5)
 => [2,3] => 10100 => 11001 => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => 11110 => 11111 => 5
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => 11110 => 11111 => 5
([(2,3),(2,4),(3,4)],5)
 => [2,3] => 10100 => 11001 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => 11111 => 11111 => 5
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => 11110 => 11111 => 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => 11101 => 11110 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => 11010 => 11101 => 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 11111 => 11111 => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => 10110 => 11011 => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 11111 => 11111 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 11111 => 11111 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => 11101 => 11110 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 11010 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => 11111 => 11111 => 5
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => 10110 => 11011 => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 11111 => 11111 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => 11011 => 11101 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => 10101 => 11010 => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 11111 => 11111 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 11111 => 11111 => 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 11111 => 11111 => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => 10010 => 10101 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => 11011 => 11101 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 10111 => 11011 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => 11111 => 11111 => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => 10101 => 11010 => 3
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Matching statistic: St001315
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001315: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1] => ([],1)
 => 1
([],2)
 => [2] => [1,1] => ([(0,1)],2)
 => 2
([(0,1)],2)
 => [1,1] => [2] => ([],2)
 => 2
([],3)
 => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([(1,2)],3)
 => [1,2] => [1,2] => ([(1,2)],3)
 => 3
([(0,2),(1,2)],3)
 => [1,1,1] => [3] => ([],3)
 => 3
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2
([],4)
 => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(2,3)],4)
 => [1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
([(1,3),(2,3)],4)
 => [1,1,2] => [1,3] => ([(2,3)],4)
 => 4
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => 3
([(0,3),(1,2)],4)
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [4] => ([],4)
 => 4
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [4] => ([],4)
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([],5)
 => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(3,4)],5)
 => [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(2,4),(3,4)],5)
 => [1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 4
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(1,4),(2,3)],5)
 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [1,4] => ([(3,4)],5)
 => 5
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,4] => ([(3,4)],5)
 => 5
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [5] => ([],5)
 => 5
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,4] => ([(3,4)],5)
 => 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => ([],5)
 => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => ([],5)
 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => ([],5)
 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [5] => ([],5)
 => 5
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => ([],5)
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => ([],5)
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => ([],5)
 => 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => ([],5)
 => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [5] => ([],5)
 => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
Description
The dissociation number of a graph.
Matching statistic: St001526
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001526: 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)
 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 2
([(0,1)],2)
 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 2
([],3)
 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2
([(1,2)],3)
 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 3
([(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 2
([(2,3)],4)
 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3
([(1,3),(2,3)],4)
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 3
([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2
([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 2
([(3,4)],5)
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3
([(2,4),(3,4)],5)
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3
([(1,4),(2,3)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 5
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 5
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3
Description
The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001499
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 98%●distinct values known / distinct values provided: 80%
Values
([],1)
 => [1] => [1,0]
 => [1,0]
 => ? = 1 - 1
([],2)
 => [2] => [1,1,0,0]
 => [1,1,0,0]
 => 1 = 2 - 1
([(0,1)],2)
 => [1,1] => [1,0,1,0]
 => [1,0,1,0]
 => 1 = 2 - 1
([],3)
 => [3] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1 = 2 - 1
([(1,2)],3)
 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
([(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
([(2,3)],4)
 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
([(1,3),(2,3)],4)
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 4 - 1
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
([(3,4)],5)
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 3 - 1
([(2,4),(3,4)],5)
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 4 - 1
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 3 - 1
([(1,4),(2,3)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 5 - 1
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 5 - 1
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 5 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3 = 4 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3 = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3 = 4 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
Description
The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.
Matching statistic: St000777
(load all 6 compositions to match this statistic)
Mapping
St000777: Graphs ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Values
([],1)
 => 1
([],2)
 => ? = 2
([(0,1)],2)
 => 2
([],3)
 => ? = 2
([(1,2)],3)
 => ? = 3
([(0,2),(1,2)],3)
 => 3
([(0,1),(0,2),(1,2)],3)
 => 2
([],4)
 => ? = 2
([(2,3)],4)
 => ? = 3
([(1,3),(2,3)],4)
 => ? = 4
([(0,3),(1,3),(2,3)],4)
 => 3
([(0,3),(1,2)],4)
 => ? = 3
([(0,3),(1,2),(2,3)],4)
 => 4
([(1,2),(1,3),(2,3)],4)
 => ? = 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([],5)
 => ? = 2
([(3,4)],5)
 => ? = 3
([(2,4),(3,4)],5)
 => ? = 4
([(1,4),(2,4),(3,4)],5)
 => ? = 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(1,4),(2,3)],5)
 => ? = 3
([(1,4),(2,3),(3,4)],5)
 => ? = 5
([(0,1),(2,4),(3,4)],5)
 => ? = 5
([(2,3),(2,4),(3,4)],5)
 => ? = 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => 5
([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => 5
([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000454
Mapping
Mp00203: Graphs —cone⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([(0,1)],2)
 => [1,1] => ([(0,1)],2)
 => 1
([],2)
 => ([(0,2),(1,2)],3)
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => ([(0,2),(1,2)],3)
 => ? = 2
([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2
([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 2
([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(1,2),(1,3),(2,3)],4)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
([(1,4),(2,3)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
([(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
([(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),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
([(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),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000259
(load all 2 compositions to match this statistic)
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 40%
Values
([],1)
 => [1] => [1] => ([],1)
 => 0 = 1 - 1
([],2)
 => [2] => [2] => ([],2)
 => ? = 2 - 1
([(0,1)],2)
 => [1,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
([],3)
 => [3] => [3] => ([],3)
 => ? = 2 - 1
([(1,2)],3)
 => [2,1] => [1,2] => ([(1,2)],3)
 => ? = 3 - 1
([(0,2),(1,2)],3)
 => [2,1] => [1,2] => ([(1,2)],3)
 => ? = 3 - 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([],4)
 => [4] => [4] => ([],4)
 => ? = 2 - 1
([(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => ? = 3 - 1
([(1,3),(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => ? = 4 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => ? = 3 - 1
([(0,3),(1,2)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 3 - 1
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 4 - 1
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 4 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 3 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
([],5)
 => [5] => [5] => ([],5)
 => ? = 2 - 1
([(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => ? = 3 - 1
([(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => ? = 4 - 1
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => ? = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => ? = 3 - 1
([(1,4),(2,3)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 3 - 1
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 5 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 5 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 5 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 4 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 5 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 5 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
Description
The diameter of a connected graph. This is the greatest distance between any pair of vertices.
Matching statistic: St000260
(load all 2 compositions to match this statistic)
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 40%
Values
([],1)
 => [1] => [1] => ([],1)
 => 0 = 1 - 1
([],2)
 => [2] => [2] => ([],2)
 => ? = 2 - 1
([(0,1)],2)
 => [1,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
([],3)
 => [3] => [3] => ([],3)
 => ? = 2 - 1
([(1,2)],3)
 => [2,1] => [1,2] => ([(1,2)],3)
 => ? = 3 - 1
([(0,2),(1,2)],3)
 => [2,1] => [1,2] => ([(1,2)],3)
 => ? = 3 - 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
([],4)
 => [4] => [4] => ([],4)
 => ? = 2 - 1
([(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => ? = 3 - 1
([(1,3),(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => ? = 4 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => ? = 3 - 1
([(0,3),(1,2)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 3 - 1
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 4 - 1
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 4 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 3 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
([],5)
 => [5] => [5] => ([],5)
 => ? = 2 - 1
([(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => ? = 3 - 1
([(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => ? = 4 - 1
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => ? = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => ? = 3 - 1
([(1,4),(2,3)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 3 - 1
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 5 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 5 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 5 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 4 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 5 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 5 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
Description
The radius of a connected graph. This is the minimum eccentricity of any vertex.