- FindStat

Identifier

Mp00250: Graphs —clique graph⟶ Graphs
St000267: Graphs ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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) => 1

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

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

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

([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 16

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

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

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

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

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

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

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

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 11

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

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

([(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) => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 75

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

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

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

([(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) => 5

([(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) => 3

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

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

([(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) => 16

([(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) => 1

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

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

([(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) => 16

([(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) => 125

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

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

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

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

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

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

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

([(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) => 16

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

([(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) => 16

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

([(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) => 11

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

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

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

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 9

([(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) => 56

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

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

([(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) => 16

([(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) => 75

([(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) => 30

([(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) => 361

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

([(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) => 8

([(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) => 16

([(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) => 16

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

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

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

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

([(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) => 1

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

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

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

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

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

([(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) => 30

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

([(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) => 14

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

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Description

The number of maximal spanning forests contained in a graph.
A maximal spanning forest in a graph is a maximal acyclic subgraph. In other words, a spanning forest is a union of spanning trees in all connected components. See also [1] for this and further definitions.
For connected graphs, this is the same as St000096The number of spanning trees of a graph..

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 $G$.