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Identifier

St000267: Graphs ⟶ ℤ

Values

([],1) => 1

([],2) => 1

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

([],3) => 1

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

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

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

([],4) => 1

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

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

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

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

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

([(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) => 4

([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 8

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

([],5) => 1

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

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

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

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

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

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

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

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

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

([(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) => 4

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

([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 8

([(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) => 8

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 12

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 20

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

([(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) => 9

([(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) => 11

([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 21

([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 8

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

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

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 40

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 24

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 45

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

([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 125

([],6) => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8

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

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

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

([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8

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

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 12

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

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 12

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 20

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 20

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 32

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 48

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

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

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

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

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

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

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

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

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

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

([(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) => 4

([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 11

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

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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..

References

[1] wikipedia:Spanning_tree#Spanning_forests
[2] wikipedia:Tutte polynomial#Individual_points

Code

def statistic(g):
    return g.tutte_polynomial().subs(x=1,y=1)

Created

Jul 28, 2015 at 09:20 by Martin Rubey

Updated

Dec 23, 2020 at 12:29 by Martin Rubey