A classical matrix-tree theorem expresses the determinant of some matrix constructed from a graph (principal minor of the Laplacian) as a sum over all spanning trees of the graph. There are generalizations of this theorem to hypergraphs or simplicial complexes [MV, DKM]. Some version of this theorem provides a formula for the first non-zero coefficient of the Conway polynomial of a (virtual) li...

This paper introduces a generalization of the Tutte polynomial [14] that is defined for signed graphs. A signed graph is a graph whose edges are each labelled with a sign (+l or 1). The generalized polynomial will be denoted Q[G] = Q[G](A, B, d). Here G is the signed graph, and the letters A, B, d denote three independent polynomial variables. The polynomial Q[G] can be specialized to the Tutte...

We look for graph polynomials which satisfy recurrence relations on three kinds of edge elimination: edge deletion, edge contraction and deletion of edges together with their end points. Like in the case of deletion and contraction only (W. Tutte, 1954), it turns out that there is a most general polynomial satisfying such recurrence relations, which we call ξ(G, x, y, z). We show that the new p...

We construct an “orbital Tutte polynomial” associated with a dual pair M and M∗ of matrices over a principal ideal domain R and a group G of automorphisms of the row spaces of the matrices. The polynomial has two sequences of variables, each sequence indexed by associate classes of elements of R. In the case where M is the signed vertex-edge incidence matrix of a graph Γ over the ring of intege...

The Tutte polynomial is one of the most important and most useful invariants of a graph. It was discovered as a two variable generalization of the chromatic polynomial [15, 16], and has been studied in literally hundreds of papers, in part due to its connections to various fields ranging from Enumerative Combinatorics to Knot Theory, from Statistical Physics to Computer Science. We refer the re...

Let G be a graph without loops or bridges and a, b be positive real numbers with b ≥ a(a + 2). We show that the Tutte polynomial of G satisfies the inequality TG(b, 0)TG(0, b) ≥ TG(a, a). Our result was inspired by a conjecture of Merino and Welsh that TG(1, 1) ≤ max{TG(2, 0), TG(0, 2)}.