Your data matches 223 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000081
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 0
[1,2] => ([],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([],1)
 => 0
[1,2,3] => ([],3)
 => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => ([],2)
 => 0
[2,1,3] => ([(1,2)],3)
 => ([],2)
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0
[1,2,3,4] => ([],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => ([],3)
 => 0
[1,3,2,4] => ([(2,3)],4)
 => ([],3)
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[2,1,3,4] => ([(2,3)],4)
 => ([],3)
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => ([],4)
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => ([],4)
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,3,2,4,5] => ([(3,4)],5)
 => ([],4)
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[2,1,3,4,5] => ([(3,4)],5)
 => ([],4)
 => 0
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
Description
The number of edges of a graph.
Matching statistic: St000171
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 0
[1,2] => ([],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([],1)
 => 0
[1,2,3] => ([],3)
 => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => ([],2)
 => 0
[2,1,3] => ([(1,2)],3)
 => ([],2)
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0
[1,2,3,4] => ([],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => ([],3)
 => 0
[1,3,2,4] => ([(2,3)],4)
 => ([],3)
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[2,1,3,4] => ([(2,3)],4)
 => ([],3)
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => ([],4)
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => ([],4)
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,3,2,4,5] => ([(3,4)],5)
 => ([],4)
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[2,1,3,4,5] => ([(3,4)],5)
 => ([],4)
 => 0
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
Description
The degree of the graph. This is the maximal vertex degree of a graph.
Matching statistic: St000312
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000312: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 0
[1,2] => ([],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([],1)
 => 0
[1,2,3] => ([],3)
 => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => ([],2)
 => 0
[2,1,3] => ([(1,2)],3)
 => ([],2)
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0
[1,2,3,4] => ([],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => ([],3)
 => 0
[1,3,2,4] => ([(2,3)],4)
 => ([],3)
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[2,1,3,4] => ([(2,3)],4)
 => ([],3)
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => ([],4)
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => ([],4)
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,3,2,4,5] => ([(3,4)],5)
 => ([],4)
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[2,1,3,4,5] => ([(3,4)],5)
 => ([],4)
 => 0
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
Description
The number of leaves in a graph. That is, the number of vertices of a graph that have degree 1.
Matching statistic: St000422
(load all 2 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 0
[1,2] => ([],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([],1)
 => 0
[1,2,3] => ([],3)
 => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => ([],2)
 => 0
[2,1,3] => ([(1,2)],3)
 => ([],2)
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0
[1,2,3,4] => ([],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => ([],3)
 => 0
[1,3,2,4] => ([(2,3)],4)
 => ([],3)
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[2,1,3,4] => ([(2,3)],4)
 => ([],3)
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => ([],4)
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => ([],4)
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,3,2,4,5] => ([(3,4)],5)
 => ([],4)
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[2,1,3,4,5] => ([(3,4)],5)
 => ([],4)
 => 0
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
Description
The energy of a graph, if it is integral. The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3]. The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St000987
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 0
[1,2] => ([],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([],1)
 => 0
[1,2,3] => ([],3)
 => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => ([],2)
 => 0
[2,1,3] => ([(1,2)],3)
 => ([],2)
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0
[1,2,3,4] => ([],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => ([],3)
 => 0
[1,3,2,4] => ([(2,3)],4)
 => ([],3)
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[2,1,3,4] => ([(2,3)],4)
 => ([],3)
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => ([],4)
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => ([],4)
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,3,2,4,5] => ([(3,4)],5)
 => ([],4)
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[2,1,3,4,5] => ([(3,4)],5)
 => ([],4)
 => 0
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
Description
The number of positive eigenvalues of the Laplacian matrix of the graph. This is the number of vertices minus the number of connected components of the graph.
Matching statistic: St001307
(load all 8 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St001307: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 0
[1,2] => ([],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([],1)
 => 0
[1,2,3] => ([],3)
 => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => ([],2)
 => 0
[2,1,3] => ([(1,2)],3)
 => ([],2)
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0
[1,2,3,4] => ([],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => ([],3)
 => 0
[1,3,2,4] => ([(2,3)],4)
 => ([],3)
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[2,1,3,4] => ([(2,3)],4)
 => ([],3)
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => ([],4)
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => ([],4)
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,3,2,4,5] => ([(3,4)],5)
 => ([],4)
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[2,1,3,4,5] => ([(3,4)],5)
 => ([],4)
 => 0
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
Description
The number of induced stars on four vertices in a graph.
Matching statistic: St001479
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St001479: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 0
[1,2] => ([],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([],1)
 => 0
[1,2,3] => ([],3)
 => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => ([],2)
 => 0
[2,1,3] => ([(1,2)],3)
 => ([],2)
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0
[1,2,3,4] => ([],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => ([],3)
 => 0
[1,3,2,4] => ([(2,3)],4)
 => ([],3)
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[2,1,3,4] => ([(2,3)],4)
 => ([],3)
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => ([],4)
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => ([],4)
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,3,2,4,5] => ([(3,4)],5)
 => ([],4)
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[2,1,3,4,5] => ([(3,4)],5)
 => ([],4)
 => 0
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
Description
The number of bridges of a graph. A bridge is an edge whose removal increases the number of connected components of the graph.
Matching statistic: St001826
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St001826: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 0
[1,2] => ([],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([],1)
 => 0
[1,2,3] => ([],3)
 => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => ([],2)
 => 0
[2,1,3] => ([(1,2)],3)
 => ([],2)
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 0
[1,2,3,4] => ([],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => ([],3)
 => 0
[1,3,2,4] => ([(2,3)],4)
 => ([],3)
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[2,1,3,4] => ([(2,3)],4)
 => ([],3)
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 0
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => ([],4)
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => ([],4)
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,3,2,4,5] => ([(3,4)],5)
 => ([],4)
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[2,1,3,4,5] => ([(3,4)],5)
 => ([],4)
 => 0
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([],3)
 => 0
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 0
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 0
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
[4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 0
[4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 0
Description
The maximal number of leaves on a vertex of a graph.
Matching statistic: St000468
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000468: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 1 = 0 + 1
[1,2] => ([],2)
 => ([],2)
 => 1 = 0 + 1
[2,1] => ([(0,1)],2)
 => ([],1)
 => 1 = 0 + 1
[1,2,3] => ([],3)
 => ([],3)
 => 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
 => ([],2)
 => 1 = 0 + 1
[2,1,3] => ([(1,2)],3)
 => ([],2)
 => 1 = 0 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1 = 0 + 1
[1,2,3,4] => ([],4)
 => ([],4)
 => 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
 => ([],3)
 => 1 = 0 + 1
[1,3,2,4] => ([(2,3)],4)
 => ([],3)
 => 1 = 0 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 1 = 0 + 1
[2,1,3,4] => ([(2,3)],4)
 => ([],3)
 => 1 = 0 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => 1 = 0 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 1 = 0 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => 1 = 0 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1 = 0 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1 = 0 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1 = 0 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1 = 0 + 1
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
 => ([],4)
 => 1 = 0 + 1
[1,2,4,3,5] => ([(3,4)],5)
 => ([],4)
 => 1 = 0 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 1 = 0 + 1
[1,3,2,4,5] => ([(3,4)],5)
 => ([],4)
 => 1 = 0 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => 1 = 0 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 1 = 0 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 1 = 0 + 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
[2,1,3,4,5] => ([(3,4)],5)
 => ([],4)
 => 1 = 0 + 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => 1 = 0 + 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([],3)
 => 1 = 0 + 1
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 4 + 1
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 1 = 0 + 1
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 1 = 0 + 1
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 1 = 0 + 1
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 1 = 0 + 1
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
[4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
[4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
[4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
Description
The Hosoya index of a graph. This is the total number of matchings in the graph.
Matching statistic: St001674
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St001674: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 1 = 0 + 1
[1,2] => ([],2)
 => ([],2)
 => 1 = 0 + 1
[2,1] => ([(0,1)],2)
 => ([],1)
 => 1 = 0 + 1
[1,2,3] => ([],3)
 => ([],3)
 => 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
 => ([],2)
 => 1 = 0 + 1
[2,1,3] => ([(1,2)],3)
 => ([],2)
 => 1 = 0 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1 = 0 + 1
[1,2,3,4] => ([],4)
 => ([],4)
 => 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
 => ([],3)
 => 1 = 0 + 1
[1,3,2,4] => ([(2,3)],4)
 => ([],3)
 => 1 = 0 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 1 = 0 + 1
[2,1,3,4] => ([(2,3)],4)
 => ([],3)
 => 1 = 0 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => 1 = 0 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => 1 = 0 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => 1 = 0 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1 = 0 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1 = 0 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1 = 0 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1 = 0 + 1
[1,2,3,4,5] => ([],5)
 => ([],5)
 => 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
 => ([],4)
 => 1 = 0 + 1
[1,2,4,3,5] => ([(3,4)],5)
 => ([],4)
 => 1 = 0 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 1 = 0 + 1
[1,3,2,4,5] => ([(3,4)],5)
 => ([],4)
 => 1 = 0 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => 1 = 0 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 1 = 0 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 1 = 0 + 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
[2,1,3,4,5] => ([(3,4)],5)
 => ([],4)
 => 1 = 0 + 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => 1 = 0 + 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([],3)
 => 1 = 0 + 1
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 4 + 1
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => 1 = 0 + 1
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => 1 = 0 + 1
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 1 = 0 + 1
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 1 = 0 + 1
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
[4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
[4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => 1 = 0 + 1
[4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => 1 = 0 + 1
Description
The number of vertices of the largest induced star graph in the graph.
The following 213 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001725The harmonious chromatic number of a graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St000311The number of vertices of odd degree in a graph. St001117The game chromatic index of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001869The maximum cut size of a graph. St000086The number of subgraphs. St000299The number of nonisomorphic vertex-induced subtrees. St001645The pebbling number of a connected graph. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000623The number of occurrences of the pattern 52341 in a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000787The number of flips required to make a perfect matching noncrossing. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St000056The decomposition (or block) number of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000068The number of minimal elements in a poset. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001490The number of connected components of a skew partition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001846The number of elements which do not have a complement in the lattice. St000929The constant term of the character polynomial of an integer partition. St000759The smallest missing part in an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000475The number of parts equal to 1 in a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001964The interval resolution global dimension of a poset. St001820The size of the image of the pop stack sorting operator. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001845The number of join irreducibles minus the rank of a lattice. St001118The acyclic chromatic index of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001684The reduced word complexity of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001862The number of crossings of a signed permutation. St000993The multiplicity of the largest part of an integer partition. St001895The oddness of a signed permutation. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. St000225Difference between largest and smallest parts in a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001249Sum of the odd parts of a partition. St001383The BG-rank of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000781The number of proper colouring schemes of a Ferrers diagram. St001128The exponens consonantiae of a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St000456The monochromatic index of a connected graph. St001545The second Elser number of a connected graph. St000464The Schultz index of a connected graph. St000069The number of maximal elements of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001429The number of negative entries in a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. St000635The number of strictly order preserving maps of a poset into itself. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St001889The size of the connectivity set of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St000753The Grundy value for the game of Kayles on a binary word. St001866The nesting alignments of a signed permutation. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000256The number of parts from which one can substract 2 and still get an integer partition. St001770The number of facets of a certain subword complex associated with the signed permutation. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000403The Szeged index minus the Wiener index of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001305The number of induced cycles on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar 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. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001395The number of strictly unfriendly partitions of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001797The number of overfull subgraphs of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001316The domatic number of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St001624The breadth of a lattice. 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. St001095The number of non-isomorphic posets with precisely one further covering relation. St000273The domination number of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000916The packing number of a graph. 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. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St001281The normalized isoperimetric number of a graph. St000322The skewness of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St000095The number of triangles of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001578The minimal number of edges to add or remove to make a graph a line graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001871The number of triconnected components of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St000297The number of leading ones in a binary word. St001430The number of positive entries in a signed permutation. St001621The number of atoms of a lattice. St001765The number of connected components of the friends and strangers graph. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word.