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A dichromatic polynomial for weighted graphs is presented. The Kauffman bracket of a signed graph, an invariant inspired by the Jones poly- nomial of a link in three-space, is shown to be essentially an evaluation of this dichromatic polynomial, as are the homfly polynomials of certain particular types of links. 1. THE DICHROMATIC POLYNOMIAL OF A WEIGHTED GRAPH In this note we use the term graph to refer to a finite multigraph, i.e. a graph G = (V(G),E(G)) which has a finite number zz > 0 of vertices and a finite number zzz > 0 of edges, some of which may be multiple edges or loops. The zeroth Betti number p0(G) of such a graph is the number of its connected components, and the first Betti number px(G) is px(G) = m - n + p0(G). A weighted graph is a graph G together with a function w mapping E(G) into some commutative ring with unity R ; if e is an edge of G then w(e) is the weight or conductance of e . If S is any set of edges of a graph G then we denote by G : S the subgraph of G which includes all the vertices of G but only the edges in S. Using this notation, we define the dichromatic polynomial Q(G ;t,z) of a weighted graph

A dichromatic polynomial for weighted graphs and link polynomials