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Identifier

St000266: 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) => 4

([],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) => 4

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

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

([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 14

([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 38

([],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) => 4

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

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

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

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

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

([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 14

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

([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 14

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 19

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 46

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

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

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

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

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

([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 18

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

([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 14

([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 38

([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 38

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 124

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 52

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 134

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 314

([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 728

([],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) => 4

([(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) => 4

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

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

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

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

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

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

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

([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 14

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

([(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) => 5

([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 14

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

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 14

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 19

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

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 19

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 46

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 14

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 46

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 65

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 146

([(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) => 4

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

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

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

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

([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 16

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 16

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

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

([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 18

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

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Description

The number of spanning subgraphs of a graph with the same connected components.
A subgraph or factor of a graph is spanning, if it has the same vertex set [1]. The present statistic additionally requires the subgraph to have the same components. It can be obtained by evaluating the Tutte polynomial at the points $x=1$ and $y=2$, see [2,3].
By mistake, [2] refers to this statistic as the number of spanning subgraphs, which would be $2^m$, where $m$ is the number of edges. Equivalently, this would be the evaluation of the Tutte polynomial at $x=y=2$.

References

[1] wikipedia:Graph_factorization
[2] wikipedia:Tutte polynomial#Individual_points
[3] Bernardi, O. Tutte polynomial, subgraphs, orientations and sandpile model: new connections via embeddings arXiv:math/0612003

Code

def statistic(g):
    return g.tutte_polynomial().subs(x=Integer(1),y=Integer(2))

Created

Jul 28, 2015 at 09:22 by Martin Rubey

Updated

Dec 23, 2015 at 08:52 by Martin Rubey