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Your data matches 114 different statistics following compositions of up to 3 maps.
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Matching statistic: St001627
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001627: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001627: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 4
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 38
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 4
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
([],7)
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 728
([(5,6)],7)
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 38
([(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 4
([(3,6),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 4
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 4
([(2,6),(3,6),(4,5),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> [1]
=> 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(1,6),(2,5),(3,4)],7)
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 3
([(2,6),(3,5),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [1]
=> 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [1]
=> 1
Description
The number of coloured connected graphs such that the multiplicities of colours are given by a partition.
In particular, the value on the partition $(n)$ is the number of unlabelled connected graphs on $n$ vertices, [[oeis:A001349]], whereas the value on the partition $(1^n)$ is the number of labelled connected graphs [[oeis:A001187]].
Matching statistic: St001490
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 14% ●values known / values provided: 21%●distinct values known / distinct values provided: 14%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 14% ●values known / values provided: 21%●distinct values known / distinct values provided: 14%
Values
([],3)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1
([],4)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? = 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 1
([],5)
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> ? = 4
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? = 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 1
([],6)
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,1],[]]
=> ? = 38
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> ? = 4
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? = 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? = 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ? = 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? = 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? = 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? = 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? = 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? = 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? = 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? = 1
([],7)
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[4,4,4,4,1],[]]
=> ? = 728
([(5,6)],7)
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3,1],[]]
=> ? = 38
([(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> ? = 4
([(3,6),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? = 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 1
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> ? = 4
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? = 1
([(2,3),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> 1
([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> ? = 4
([(2,6),(3,6),(4,5),(5,6)],7)
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? = 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? = 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? = 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 1
([(1,6),(2,5),(3,4)],7)
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? = 3
([(2,6),(3,5),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 1
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 1
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? = 1
([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 1
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 1
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 1
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 1
([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ? = 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> ? = 4
([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3,1]]
=> ? = 1
([(2,7),(3,7),(4,6),(5,6)],8)
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 1
([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 1
([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> 1
([(1,3),(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> 1
([(1,2),(1,3),(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 1
([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 1
Description
The number of connected components of a skew partition.
Matching statistic: St000264
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Values
([],3)
=> ([],1)
=> [1] => ([],1)
=> ? = 1 + 2
([],4)
=> ([],1)
=> [1] => ([],1)
=> ? = 1 + 2
([(2,3)],4)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> ? = 1 + 2
([],5)
=> ([],1)
=> [1] => ([],1)
=> ? = 4 + 2
([(3,4)],5)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> ? = 1 + 2
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> ? = 1 + 2
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 2
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 2
([],6)
=> ([],1)
=> [1] => ([],1)
=> ? = 38 + 2
([(4,5)],6)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> ? = 4 + 2
([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> ? = 1 + 2
([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> ? = 1 + 2
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 2
([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 2
([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> ? = 1 + 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 2
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
([],7)
=> ([],1)
=> [1] => ([],1)
=> ? = 728 + 2
([(5,6)],7)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> ? = 38 + 2
([(4,6),(5,6)],7)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> ? = 4 + 2
([(3,6),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> ? = 1 + 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> ? = 1 + 2
([(3,6),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 4 + 2
([(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(2,3),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 2
([(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 4 + 2
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(2,3)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 2
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> ? = 1 + 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,4),(2,3)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 2
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,2)],3)
=> [1,2] => ([(1,2)],3)
=> ? = 1 + 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 2
([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 3 + 2
([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3)],6)
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 2
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 2
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? => ?
=> ? = 4 + 2
([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? => ?
=> ? = 1 + 2
([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? => ?
=> ? = 1 + 2
([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? => ?
=> ? = 1 + 2
([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? => ?
=> ? = 1 + 2
([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? => ?
=> ? = 1 + 2
([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? => ?
=> ? = 1 + 2
([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? => ?
=> ? = 1 + 2
([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? => ?
=> ? = 1 + 2
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001704
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
([],3)
=> ([],3)
=> ([],1)
=> ? = 1 + 1
([],4)
=> ([],4)
=> ([],1)
=> ? = 1 + 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> ? = 1 + 1
([],5)
=> ([],5)
=> ([],1)
=> ? = 4 + 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> ? = 1 + 1
([(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ? = 1 + 1
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ? = 1 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([],6)
=> ([],6)
=> ([],1)
=> ? = 38 + 1
([(4,5)],6)
=> ([],5)
=> ([],1)
=> ? = 4 + 1
([(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ? = 1 + 1
([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ([],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)
=> 2 = 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)
=> 2 = 1 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 2 = 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)
=> 2 = 1 + 1
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],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)
=> 2 = 1 + 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([],7)
=> ([],7)
=> ?
=> ? = 728 + 1
([(5,6)],7)
=> ([],6)
=> ([],1)
=> ? = 38 + 1
([(4,6),(5,6)],7)
=> ([],5)
=> ([],1)
=> ? = 4 + 1
([(3,6),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ? = 1 + 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 1 + 1
([(3,6),(4,5)],7)
=> ([],5)
=> ([],1)
=> ? = 4 + 1
([(3,6),(4,5),(5,6)],7)
=> ([],4)
=> ([],1)
=> ? = 1 + 1
([(2,3),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ? = 1 + 1
([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 + 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 1 + 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 1 + 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 1 + 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([],3)
=> ([],1)
=> ? = 1 + 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 + 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 1 + 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 + 1
([(1,6),(2,5),(3,4)],7)
=> ([],4)
=> ([],1)
=> ? = 3 + 1
([(2,6),(3,5),(4,5),(4,6)],7)
=> ([],3)
=> ([],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
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 1 + 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ?
=> ? = 1 + 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 + 1
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 + 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 + 1
([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 + 1
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 1 + 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 1 + 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 + 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ?
=> ? = 1 + 1
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 + 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 4 + 1
([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 1 + 1
([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 1 + 1
([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 1 + 1
Description
The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph.
The deck of a graph is the multiset of induced subgraphs obtained by deleting a single vertex.
The graph reconstruction conjecture states that the deck of a graph with at least three vertices determines the graph.
This statistic is only defined for graphs with at least two vertices, because there is only a single graph of the given size otherwise.
Matching statistic: St000455
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
([],3)
=> ([],3)
=> ([],1)
=> ? = 1 - 2
([],4)
=> ([],4)
=> ([],1)
=> ? = 1 - 2
([(2,3)],4)
=> ([],3)
=> ([],1)
=> ? = 1 - 2
([],5)
=> ([],5)
=> ([],1)
=> ? = 4 - 2
([(3,4)],5)
=> ([],4)
=> ([],1)
=> ? = 1 - 2
([(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ? = 1 - 2
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ? = 1 - 2
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([],6)
=> ([],6)
=> ([],1)
=> ? = 38 - 2
([(4,5)],6)
=> ([],5)
=> ([],1)
=> ? = 4 - 2
([(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ? = 1 - 2
([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ? = 1 - 2
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ? = 1 - 2
([(2,5),(3,4),(4,5)],6)
=> ([],3)
=> ([],1)
=> ? = 1 - 2
([(1,2),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ? = 1 - 2
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> -1 = 1 - 2
([(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)
=> -1 = 1 - 2
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ? = 1 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 1 - 2
([],7)
=> ([],7)
=> ?
=> ? = 728 - 2
([(5,6)],7)
=> ([],6)
=> ([],1)
=> ? = 38 - 2
([(4,6),(5,6)],7)
=> ([],5)
=> ([],1)
=> ? = 4 - 2
([(3,6),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ? = 1 - 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 1 - 2
([(3,6),(4,5)],7)
=> ([],5)
=> ([],1)
=> ? = 4 - 2
([(3,6),(4,5),(5,6)],7)
=> ([],4)
=> ([],1)
=> ? = 1 - 2
([(2,3),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ? = 1 - 2
([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 4 - 2
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 1 - 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 1 - 2
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 1 - 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([],3)
=> ([],1)
=> ? = 1 - 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> -1 = 1 - 2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 - 2
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 1 - 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 - 2
([(1,6),(2,5),(3,4)],7)
=> ([],4)
=> ([],1)
=> ? = 3 - 2
([(2,6),(3,5),(4,5),(4,6)],7)
=> ([],3)
=> ([],1)
=> ? = 1 - 2
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 1 - 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ? = 1 - 2
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 1 - 2
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ?
=> ? = 1 - 2
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 - 2
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 - 2
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> -1 = 1 - 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 - 2
([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 1 - 2
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 - 2
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 1 - 2
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 1 - 2
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 - 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ?
=> ? = 1 - 2
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 1 - 2
([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 - 2
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 4 - 2
([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 1 - 2
([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 1 - 2
([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 1 - 2
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000379
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([],3)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([],4)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([],5)
=> ([],5)
=> ([],1)
=> ([],1)
=> ? = 4 - 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ([],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)
=> 0 = 1 - 1
([],6)
=> ([],6)
=> ([],1)
=> ([],1)
=> ? = 38 - 1
([(4,5)],6)
=> ([],5)
=> ([],1)
=> ([],1)
=> ? = 4 - 1
([(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,5),(3,4),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(1,2),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],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)
=> 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)
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 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)
=> 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ([],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)
=> 0 = 1 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 1 - 1
([],7)
=> ([],7)
=> ?
=> ?
=> ? = 728 - 1
([(5,6)],7)
=> ([],6)
=> ([],1)
=> ([],1)
=> ? = 38 - 1
([(4,6),(5,6)],7)
=> ([],5)
=> ([],1)
=> ([],1)
=> ? = 4 - 1
([(3,6),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(3,6),(4,5)],7)
=> ([],5)
=> ([],1)
=> ([],1)
=> ? = 4 - 1
([(3,6),(4,5),(5,6)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,3),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 4 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(1,6),(2,5),(3,4)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 3 - 1
([(2,6),(3,5),(4,5),(4,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 1 - 1
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 1 - 1
([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 4 - 1
([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 1 - 1
([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 1 - 1
([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 1 - 1
Description
The number of Hamiltonian cycles in a graph.
A Hamiltonian cycle in a graph $G$ is a subgraph (this is, a subset of the edges) that is a cycle which contains every vertex of $G$.
Since it is unclear whether the graph on one vertex is Hamiltonian, the statistic is undefined for this graph.
Matching statistic: St000699
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],3)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([],4)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([],5)
=> ([],5)
=> ([],1)
=> ([],1)
=> ? = 4 - 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ([],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)
=> 0 = 1 - 1
([],6)
=> ([],6)
=> ([],1)
=> ([],1)
=> ? = 38 - 1
([(4,5)],6)
=> ([],5)
=> ([],1)
=> ([],1)
=> ? = 4 - 1
([(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,5),(3,4),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(1,2),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],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)
=> 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)
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 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)
=> 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ([],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)
=> 0 = 1 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 1 - 1
([],7)
=> ([],7)
=> ?
=> ?
=> ? = 728 - 1
([(5,6)],7)
=> ([],6)
=> ([],1)
=> ([],1)
=> ? = 38 - 1
([(4,6),(5,6)],7)
=> ([],5)
=> ([],1)
=> ([],1)
=> ? = 4 - 1
([(3,6),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(3,6),(4,5)],7)
=> ([],5)
=> ([],1)
=> ([],1)
=> ? = 4 - 1
([(3,6),(4,5),(5,6)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,3),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 4 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(1,6),(2,5),(3,4)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 3 - 1
([(2,6),(3,5),(4,5),(4,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 1 - 1
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 1 - 1
([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 4 - 1
([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 1 - 1
([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 1 - 1
([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 1 - 1
Description
The toughness times the least common multiple of 1,...,n-1 of a non-complete graph.
A graph $G$ is $t$-tough if $G$ cannot be split into $k$ different connected components by the removal of fewer than $tk$ vertices for all integers $k>1$.
The toughness of $G$ is the maximal number $t$ such that $G$ is $t$-tough. It is a rational number except for the complete graph, where it is infinity. The toughness of a disconnected graph is zero.
This statistic is the toughness multiplied by the least common multiple of $1,\dots,n-1$, where $n$ is the number of vertices of $G$.
Matching statistic: St001281
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([],3)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([],4)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([],5)
=> ([],5)
=> ([],1)
=> ([],1)
=> ? = 4 - 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ([],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)
=> 0 = 1 - 1
([],6)
=> ([],6)
=> ([],1)
=> ([],1)
=> ? = 38 - 1
([(4,5)],6)
=> ([],5)
=> ([],1)
=> ([],1)
=> ? = 4 - 1
([(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,5),(3,4),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(1,2),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],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)
=> 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)
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 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)
=> 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ([],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)
=> 0 = 1 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 1 - 1
([],7)
=> ([],7)
=> ?
=> ?
=> ? = 728 - 1
([(5,6)],7)
=> ([],6)
=> ([],1)
=> ([],1)
=> ? = 38 - 1
([(4,6),(5,6)],7)
=> ([],5)
=> ([],1)
=> ([],1)
=> ? = 4 - 1
([(3,6),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(3,6),(4,5)],7)
=> ([],5)
=> ([],1)
=> ([],1)
=> ? = 4 - 1
([(3,6),(4,5),(5,6)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,3),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 4 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(1,6),(2,5),(3,4)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 3 - 1
([(2,6),(3,5),(4,5),(4,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 1 - 1
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 1 - 1
([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 4 - 1
([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 1 - 1
([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 1 - 1
([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 1 - 1
Description
The normalized isoperimetric number of a graph.
The isoperimetric number, or Cheeger constant, of a graph $G$ is
$$
i(G) = \min\left\{\frac{|\partial A|}{|A|}\ : \ A\subseteq V(G), 0 < |A|\leq |V(G)|/2\right\},
$$
where
$$
\partial A := \{(x, y)\in E(G)\ : \ x\in A, y\in V(G)\setminus A \}.
$$
This statistic is $i(G)\cdot\lfloor n/2\rfloor$.
Matching statistic: St001592
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([],3)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([],4)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([],5)
=> ([],5)
=> ([],1)
=> ([],1)
=> ? = 4 - 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> ([],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)
=> 0 = 1 - 1
([],6)
=> ([],6)
=> ([],1)
=> ([],1)
=> ? = 38 - 1
([(4,5)],6)
=> ([],5)
=> ([],1)
=> ([],1)
=> ? = 4 - 1
([(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,5),(3,4),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(1,2),(3,5),(4,5)],6)
=> ([],3)
=> ([],1)
=> ([],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)
=> 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)
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 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)
=> 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([],1)
=> ([],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)
=> 0 = 1 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 1 - 1
([],7)
=> ([],7)
=> ?
=> ?
=> ? = 728 - 1
([(5,6)],7)
=> ([],6)
=> ([],1)
=> ([],1)
=> ? = 38 - 1
([(4,6),(5,6)],7)
=> ([],5)
=> ([],1)
=> ([],1)
=> ? = 4 - 1
([(3,6),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(3,6),(4,5)],7)
=> ([],5)
=> ([],1)
=> ([],1)
=> ? = 4 - 1
([(3,6),(4,5),(5,6)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,3),(4,6),(5,6)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 4 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(1,6),(2,5),(3,4)],7)
=> ([],4)
=> ([],1)
=> ([],1)
=> ? = 3 - 1
([(2,6),(3,5),(4,5),(4,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 1 - 1
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 1 - 1
([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 1 - 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 4 - 1
([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 1 - 1
([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 1 - 1
([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 1 - 1
Description
The maximal number of simple paths between any two different vertices of a graph.
Matching statistic: St001545
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 38 - 1
([(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
([(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(2,5),(3,4)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
([],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 728 - 1
([(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 38 - 1
([(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4 - 1
([(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(3,6),(4,5)],7)
=> ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 4 - 1
([(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(2,3),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,3),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,2),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(1,6),(2,5),(3,4)],7)
=> ([(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 3 - 1
([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(0,7),(1,2),(1,7),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(0,3),(0,7),(1,2),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,3),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,5),(2,7),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,2),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,7),(1,2),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,2),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,7),(1,2),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,2),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 4 - 1
Description
The second Elser number of a connected graph.
For a connected graph $G$ the $k$-th Elser number is
$$
els_k(G) = (-1)^{|V(G)|+1} \sum_N (-1)^{|E(N)|} |V(N)|^k
$$
where the sum is over all nuclei of $G$, that is, the connected subgraphs of $G$ whose vertex set is a vertex cover of $G$.
It is clear that this number is even. It was shown in [1] that it is non-negative.
The following 104 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001570The minimal number of edges to add to make a graph Hamiltonian. St001703The villainy of a graph. St001060The distinguishing index of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000456The monochromatic index of a connected graph. St001118The acyclic chromatic index of a graph. St000464The Schultz index of a connected graph. St000287The number of connected components of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000535The rank-width of a graph. St000553The number of blocks of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000916The packing number of a graph. St000917The open packing number of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001272The number of graphs with the same degree sequence. St001282The number of graphs with the same chromatic polynomial. St001333The cardinality of a minimal edge-isolating set of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001765The number of connected components of the friends and strangers graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000258The burning number of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000311The number of vertices of odd degree in a graph. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000370The genus of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000918The 2-limited packing number of a graph. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001261The Castelnuovo-Mumford regularity of a graph. St001306The number of induced paths on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001315The dissociation number of a graph. St001323The independence gap of a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001367The smallest number which does not occur as degree of a vertex in a graph. St001395The number of strictly unfriendly partitions of a graph. St001479The number of bridges of a graph. St001521Half the total irregularity of a graph. St001522The total irregularity of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001689The number of celebrities in a graph. St001691The number of kings in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001708The number of pairs of vertices of different degree in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001793The difference between the clique number and the chromatic number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St000172The Grundy number of a graph. St000286The number of connected components of the complement of a graph. St000363The number of minimal vertex covers of a graph. St000469The distinguishing number of a graph. St000722The number of different neighbourhoods in a graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001458The rank of the adjacency matrix of a graph. St001494The Alon-Tarsi number of a graph. St001581The achromatic number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001109The number of proper colourings of a graph with as few colours as possible.
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