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Identifier

St001071: Graphs ⟶ ℤ

Values

([],1) => 0

([],2) => 0

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

([],3) => 0

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

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

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

([],4) => 0

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

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

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

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

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

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

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

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

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

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

([],5) => 0

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

([(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) => 1

([(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) => 1

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

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

([(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) => 1

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

([(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) => 1

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

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

([(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) => 2

([(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) => 2

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

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

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

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

([],6) => 0

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

([(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) => 1

([(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) => 1

([(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) => 1

([(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) => 1

([(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) => 1

([(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) => 1

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

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

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

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

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

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

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

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

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

([(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) => 0

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

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

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

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

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Description

The beta invariant of the graph.
The beta invariant was introduced by Crapo [1] for matroids. For graphs with $n$ vertices the beta invariant is
$$\beta(G) = (-1)^{n-c}\sum_{S\subseteq E} (-1)^{|S|} (n-c(S)),$$
where $c(S)$ is the number of connected components of the subgraph of $G$ with edge set $S$.
For graphs with at least one edge the beta invariant equals the absolute value of the derivative of the chromatic polynomial at $1$. [2]
The beta invariant also coincides with the coefficient of the monomial $x$, and also with the coefficient of the monomial $y$, of the Tutte polynomial.

References

[1] Crapo, H. H. A higher invariant for matroids MathSciNet:0215744
[2] Benashski, J. K., Martin, R. R., Moore, J. T., Traldi, L. On the β-invariant for graphs MathSciNet:1369311

Code

def statistic(G):
    if G.num_verts() == G.connected_components_number():
        return 0
    return abs(G.chromatic_polynomial().diff().subs(x=1))

Created

Jan 04, 2018 at 17:15 by Martin Rubey

Updated

Sep 13, 2023 at 14:55 by Martin Rubey