- FindStat

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Identifier

St000274: Graphs ⟶ ℤ

Values

([],1) => 0

([],2) => 0

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

([],3) => 0

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

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

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

([],4) => 0

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

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

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

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

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

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

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

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

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

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

([],5) => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([],6) => 0

([(4,5)],6) => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 4$

$F_{4} = 5 + 3\ q + 2\ q^{2} + q^{3}$

$F_{5} = 34$

$F_{6} = 55 + 23\ q + 29\ q^{2} + 15\ q^{3} + 13\ q^{4} + 5\ q^{5} + 8\ q^{6} + 2\ q^{7} + 2\ q^{8} + q^{9} + q^{10} + q^{12} + q^{15}$

Description

The number of perfect matchings of a graph.
A matching of a graph

$G$ is a subset

$F \subset E(G)$ such that no two edges in

$F$ share a vertex in common. A perfect matching

$F'$ is then a matching such that every vertex in

$V(G)$ is incident with exactly one edge in

$F'$ .

References

[1] wikipedia:Matching polynomial

Code

def statistic(g):
    return abs(g.matching_polynomial()(0))

Created

Jul 28, 2015 at 18:52 by Martin Rubey

Updated

Dec 17, 2015 at 22:58 by Matthew Donahue