- FindStat

Identifier

Mp00250: Graphs —clique graph⟶ Graphs
St001463: Graphs ⟶ ℤ

Values

([],1) => ([],1) => 1

([],2) => ([],2) => 2

([(0,1)],2) => ([],1) => 1

([],3) => ([],3) => 3

([(1,2)],3) => ([],2) => 2

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

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

([],4) => ([],4) => 4

([(2,3)],4) => ([],3) => 3

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

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

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

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

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

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

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

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

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

([],5) => ([],5) => 5

([(3,4)],5) => ([],4) => 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([],6) => ([],6) => 6

([(4,5)],6) => ([],5) => 5

([(3,5),(4,5)],6) => ([(3,4)],5) => 4

([(2,5),(3,5),(4,5)],6) => ([(2,3),(2,4),(3,4)],5) => 3

([(1,5),(2,5),(3,5),(4,5)],6) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2

([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1

([(2,5),(3,4)],6) => ([],4) => 4

([(2,5),(3,4),(4,5)],6) => ([(2,4),(3,4)],5) => 5

([(1,2),(3,5),(4,5)],6) => ([(2,3)],4) => 3

([(3,4),(3,5),(4,5)],6) => ([],4) => 4

([(1,5),(2,5),(3,4),(4,5)],6) => ([(1,4),(2,3),(2,4),(3,4)],5) => 2

([(0,1),(2,5),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => 2

([(2,5),(3,4),(3,5),(4,5)],6) => ([(2,3)],4) => 3

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1

([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => 2

([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1

([(2,4),(2,5),(3,4),(3,5)],6) => ([(2,4),(2,5),(3,4),(3,5)],6) => 6

([(0,5),(1,5),(2,4),(3,4)],6) => ([(0,3),(1,2)],4) => 1

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 4

([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 1

([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(2,3)],4) => 3

([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,3),(2,3)],4) => 4

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 1

([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3

([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => 2

([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,3),(1,2),(1,3),(2,3)],4) => 1

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5)],7) => 4

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6) => 3

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1),(0,2),(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => 2

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1

([(0,5),(1,4),(2,3)],6) => ([],3) => 3

([(1,5),(2,4),(3,4),(3,5)],6) => ([(1,4),(2,3),(3,4)],5) => 2

([(0,1),(2,5),(3,4),(4,5)],6) => ([(1,3),(2,3)],4) => 4

([(1,2),(3,4),(3,5),(4,5)],6) => ([],3) => 3

([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 3

([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,3),(2,3)],4) => 4

([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2)],3) => 2

([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,3),(1,2),(1,3),(2,3)],4) => 1

([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,2)],3) => 2

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 1

([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 2

([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 1

([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,3),(1,4),(2,3),(2,4)],5) => 5

([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 1

([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,3),(2,3)],4) => 4

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Description

The number of distinct columns in the nullspace of a graph.
Let $A$ be the adjacency matrix of a graph on $n$ vertices, and $K$ a $n\times d$ matrix whose column vectors form a basis of the nullspace of $A$. Then any other matrix $K'$ whose column vectors also form a basis of the nullspace is related to $K$ by $K' = K T$ for some invertible $d\times d$ matrix $T$. Any two rows of $K$ are equal if and only if they are equal in $K'$.
The nullspace of a graph is usually written as a $d\times n$ matrix, hence the name of this statistic.

Map

clique graph

Description

The clique graph of a graph.
The clique graph of a graph $G$ has as vertex set the set of maximal cliques $G$ and an edge between vertices corresponding to cliques that intersect.
In other words, it is the intersection graph of the maximal cliques of $G$.