- FindStat

Your data matches 188 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000098
(load all 22 compositions to match this statistic)

Mapping

St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => 1
([],2)
 => 1
([(0,1)],2)
 => 2
([],3)
 => 1
([(1,2)],3)
 => 2
([(0,2),(1,2)],3)
 => 2
([(0,1),(0,2),(1,2)],3)
 => 3
([],4)
 => 1
([(2,3)],4)
 => 2
([(1,3),(2,3)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2)],4)
 => 2
([(0,3),(1,2),(2,3)],4)
 => 2
([(1,2),(1,3),(2,3)],4)
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([],5)
 => 1
([(3,4)],5)
 => 2
([(2,4),(3,4)],5)
 => 2
([(1,4),(2,4),(3,4)],5)
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
([(1,4),(2,3)],5)
 => 2
([(1,4),(2,3),(3,4)],5)
 => 2
([(0,1),(2,4),(3,4)],5)
 => 2
([(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 3

Description

The chromatic number of a graph. The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.

Matching statistic: St000172
(load all 11 compositions to match this statistic)

Mapping

Mp00154: Graphs —core⟶ Graphs
St000172: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 1
([],2)
 => ([],1)
 => 1
([(0,1)],2)
 => ([(0,1)],2)
 => 2
([],3)
 => ([],1)
 => 1
([(1,2)],3)
 => ([(0,1)],2)
 => 2
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([],4)
 => ([],1)
 => 1
([(2,3)],4)
 => ([(0,1)],2)
 => 2
([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(1,2)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 2
([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([],5)
 => ([],1)
 => 1
([(3,4)],5)
 => ([(0,1)],2)
 => 2
([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(0,1),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3

Description

The Grundy number of a graph. The Grundy number

$\Gamma(G)$ is defined to be the largest

$k$ such that

$G$ admits a greedy

$k$ -coloring. Any order of the vertices of

$G$ induces a greedy coloring by assigning to the

$i$ -th vertex in this order the smallest positive integer such that the partial coloring remains a proper coloring. In particular, we have that

$\chi(G) \leq \Gamma(G) \leq \Delta(G) + 1$ , where

$\chi(G)$ is the chromatic number of

$G$ ([[St000098]]), and where

$\Delta(G)$ is the maximal degree of a vertex of

$G$ ([[St000171]]).

Matching statistic: St000822
(load all 15 compositions to match this statistic)

Mapping

Mp00154: Graphs —core⟶ Graphs
St000822: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 1
([],2)
 => ([],1)
 => 1
([(0,1)],2)
 => ([(0,1)],2)
 => 2
([],3)
 => ([],1)
 => 1
([(1,2)],3)
 => ([(0,1)],2)
 => 2
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([],4)
 => ([],1)
 => 1
([(2,3)],4)
 => ([(0,1)],2)
 => 2
([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(1,2)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 2
([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([],5)
 => ([],1)
 => 1
([(3,4)],5)
 => ([(0,1)],2)
 => 2
([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(0,1),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3

Description

The Hadwiger number of the graph. Also known as clique contraction number, this is the size of the largest complete minor.

Matching statistic: St001116
(load all 10 compositions to match this statistic)

Mapping

Mp00154: Graphs —core⟶ Graphs
St001116: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 1
([],2)
 => ([],1)
 => 1
([(0,1)],2)
 => ([(0,1)],2)
 => 2
([],3)
 => ([],1)
 => 1
([(1,2)],3)
 => ([(0,1)],2)
 => 2
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([],4)
 => ([],1)
 => 1
([(2,3)],4)
 => ([(0,1)],2)
 => 2
([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(1,2)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 2
([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([],5)
 => ([],1)
 => 1
([(3,4)],5)
 => ([(0,1)],2)
 => 2
([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(0,1),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3

Description

The game chromatic number of a graph. Two players, Alice and Bob, take turns colouring properly any uncolored vertex of the graph. Alice begins. If it is not possible for either player to colour a vertex, then Bob wins. If the graph is completely colored, Alice wins. The game chromatic number is the smallest number of colours such that Alice has a winning strategy.

Matching statistic: St001494
(load all 15 compositions to match this statistic)

Mapping

Mp00154: Graphs —core⟶ Graphs
St001494: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 1
([],2)
 => ([],1)
 => 1
([(0,1)],2)
 => ([(0,1)],2)
 => 2
([],3)
 => ([],1)
 => 1
([(1,2)],3)
 => ([(0,1)],2)
 => 2
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([],4)
 => ([],1)
 => 1
([(2,3)],4)
 => ([(0,1)],2)
 => 2
([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(1,2)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 2
([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([],5)
 => ([],1)
 => 1
([(3,4)],5)
 => ([(0,1)],2)
 => 2
([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(0,1),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3

Description

The Alon-Tarsi number of a graph. Let

$G$ be a graph with vertices

$\{1,\dots,n\}$ and edge set

$E$ . Let

$P_G=\prod_{i < j, (i,j)\in E} x_i-x_j$ be its graph polynomial. Then the Alon-Tarsi number is the smallest number

$k$ such that

$P_G$ contains a monomial with exponents strictly less than

$k$ .

Matching statistic: St001580
(load all 11 compositions to match this statistic)

Mapping

Mp00154: Graphs —core⟶ Graphs
St001580: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 1
([],2)
 => ([],1)
 => 1
([(0,1)],2)
 => ([(0,1)],2)
 => 2
([],3)
 => ([],1)
 => 1
([(1,2)],3)
 => ([(0,1)],2)
 => 2
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([],4)
 => ([],1)
 => 1
([(2,3)],4)
 => ([(0,1)],2)
 => 2
([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(1,2)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 2
([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([],5)
 => ([],1)
 => 1
([(3,4)],5)
 => ([(0,1)],2)
 => 2
([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(0,1),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3

Description

The acyclic chromatic number of a graph. This is the smallest size of a vertex partition

$\{V_1,\dots,V_k\}$ such that each

$V_i$ is an independent set and for all

$i,j$ the subgraph inducted by

$V_i\cup V_j$ does not contain a cycle.

Matching statistic: St001581
(load all 11 compositions to match this statistic)

Mapping

Mp00154: Graphs —core⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 1
([],2)
 => ([],1)
 => 1
([(0,1)],2)
 => ([(0,1)],2)
 => 2
([],3)
 => ([],1)
 => 1
([(1,2)],3)
 => ([(0,1)],2)
 => 2
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([],4)
 => ([],1)
 => 1
([(2,3)],4)
 => ([(0,1)],2)
 => 2
([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(1,2)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 2
([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([],5)
 => ([],1)
 => 1
([(3,4)],5)
 => ([(0,1)],2)
 => 2
([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(0,1),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3

Description

The achromatic number of a graph. This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.

Matching statistic: St001670
(load all 11 compositions to match this statistic)

Mapping

Mp00154: Graphs —core⟶ Graphs
St001670: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 1
([],2)
 => ([],1)
 => 1
([(0,1)],2)
 => ([(0,1)],2)
 => 2
([],3)
 => ([],1)
 => 1
([(1,2)],3)
 => ([(0,1)],2)
 => 2
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([],4)
 => ([],1)
 => 1
([(2,3)],4)
 => ([(0,1)],2)
 => 2
([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(1,2)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 2
([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([],5)
 => ([],1)
 => 1
([(3,4)],5)
 => ([(0,1)],2)
 => 2
([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(0,1),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3

Description

The connected partition number of a graph. This is the maximal number of blocks of a set partition

$P$ of the set of vertices of a graph such that contracting each block of

$P$ to a single vertex yields a clique. Also called the pseudoachromatic number of a graph. This is the largest

$n$ such that there exists a (not necessarily proper)

$n$ -coloring of the graph so that every two distinct colors are adjacent.

Matching statistic: St000261
(load all 4 compositions to match this statistic)

Mapping

Mp00154: Graphs —core⟶ Graphs
St000261: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 0 = 1 - 1
([],2)
 => ([],1)
 => 0 = 1 - 1
([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
([],3)
 => ([],1)
 => 0 = 1 - 1
([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([],4)
 => ([],1)
 => 0 = 1 - 1
([(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([],5)
 => ([],1)
 => 0 = 1 - 1
([(3,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1

Description

The edge connectivity of a graph. This is the minimum number of edges that has to be removed to make the graph disconnected.

Matching statistic: St000262
(load all 4 compositions to match this statistic)

Mapping

Mp00154: Graphs —core⟶ Graphs
St000262: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 0 = 1 - 1
([],2)
 => ([],1)
 => 0 = 1 - 1
([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
([],3)
 => ([],1)
 => 0 = 1 - 1
([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([],4)
 => ([],1)
 => 0 = 1 - 1
([(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([],5)
 => ([],1)
 => 0 = 1 - 1
([(3,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1

Description

The vertex connectivity of a graph. For non-complete graphs, this is the minimum number of vertices that has to be removed to make the graph disconnected.

The following 178 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000272The treewidth of a graph. St000310The minimal degree of a vertex of a graph. St000536The pathwidth of a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001644The dimension of a graph. St001962The proper pathwidth of a graph. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000097The order of the largest clique of the graph. St000288The number of ones in a binary word. St000482The (zero)-forcing number of a graph. St000636The hull number of a graph. St000820The number of compositions obtained by rotating the composition. St000918The 2-limited packing number of a graph. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001202Call 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. St001203We associate to 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 Dyck path as follows: St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001315The dissociation number of a graph. St001655The general position number of a graph. St001672The restrained domination number of a graph. St001963The tree-depth of a graph. St000053The number of valleys of the Dyck path. St000306The bounce count of a Dyck path. St000362The size of a minimal vertex cover of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001108The 2-dynamic chromatic number of a graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001358The largest degree of a regular subgraph of a graph. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001746The coalition number of a graph. St001777The number of weak descents in an integer composition. St000087The number of induced subgraphs. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000228The size of a partition. St000286The number of connected components of the complement of a graph. St000363The number of minimal vertex covers of a graph. St000384The maximal part of the shifted composition of an integer partition. St000393The number of strictly increasing runs in a binary word. St000443The number of long tunnels of a Dyck path. St000469The distinguishing number of a graph. St000722The number of different neighbourhoods in a graph. St000733The row containing the largest entry of a standard tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000784The maximum of the length and the largest part of the integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000926The clique-coclique number of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001110The 3-dynamic chromatic number of a graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001286The annihilation number of a graph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001437The flex of a binary word. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001656The monophonic position number of a graph. St001675The number of parts equal to the part in the reversed composition. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St000024The number of double up and double down steps of a Dyck path. St000120The number of left tunnels of a Dyck path. St000157The number of descents of a standard tableau. St000171The degree of the graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000454The largest eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001321The number of vertices of the largest induced subforest of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001391The disjunction number of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001949The rigidity index of a graph. St001480The number of simple summands of the module J^2/J^3. St001812The biclique partition number of a graph. St001331The size of the minimal feedback vertex set. St001742The difference of the maximal and the minimal degree in a graph. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000299The number of nonisomorphic vertex-induced subtrees. St000479The Ramsey number of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001834The number of non-isomorphic minors of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001323The independence gap of a graph. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000260The radius of a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001638The book thickness of a graph. St000259The diameter of a connected graph. St001060The distinguishing index of a graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000144The pyramid weight of the Dyck path. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000264The girth of a graph, which is not a tree. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000738The first entry in the last row of a standard tableau. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001530The depth of a Dyck path. St000474Dyson's crank of a partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001875The number of simple modules with projective dimension at most 1. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001165Number of simple modules with even projective dimension in the corresponding 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

$eA$ . St001118The acyclic chromatic index of a graph. St000395The sum of the heights of the peaks of a Dyck path. St000439The position of the first down step of a Dyck path. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n-1}]$ by adding

$c_0$ to

$c_{n-1}$ . St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001267The length of the Lyndon factorization of the binary word. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001814The number of partitions interlacing the given partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000455The second largest eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001621The number of atoms of a lattice. St001624The breadth of a lattice. 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. St000776The maximal multiplicity of an eigenvalue in a graph.