9 S ep 2 00 2 A TWO - VARIABLE INTERLACE POLYNOMIAL RICHARD

Generalizing a previous one-variable " interlace polynomial " , we consider a new interlace polynomial in two variables. The polynomial can be computed in two very different ways. The first is an expansion analogous to the state space expansion of the Tutte polynomial; the differences are that our expansion is over vertex rather than edge subsets, the rank of the subset appears positively rather than subtracted from the rank of the whole, and the rank and nullity are taken over F 2 rather than R. The second computation is by a three-term reduction formula involving a graph pivot; the pivot arose previously in the study of interlacement and Euler circuits in four-regular graphs. We consider a few properties and specializations of the two-variable interlace polynomial. One specialization, the " vertex-nullity polynomial " , is the earlier one-variable inter-lace polynomial. Another, the " vertex-rank polynomial " , is also interesting. Yet another specialization of the two-variable polynomial is the partition function for independent sets in a graph. 1. The polynomial Given a graph G with vertex set V (G), for any subset S ⊂ V (G), let G[S] be the subgraph of G induced by S. Somewhat unconventionally, we shall allow the null graph with no vertices, writing G for the set of graphs including the null graph. In particular, if S is the empty set then G[S] is the null graph of rank and nullity 0. For a matrix A over F 2 , let n(A) be the nullity of A and r(A) its rank. Abusing notation slightly, for a graph G, n(G) and r(G) will denote the nullity and rank of its adjacency matrix. It is a fact from linear algebra that for a symmetric matrix A over F 2 with zero diagonal, r(A) is always even. We are ready to define our two-variable interlace polynomial q(G; x, y) of a graph G of order n as a sum of 2 n terms: