A Two-Variable Interlace Polynomial

Abstract

We introduce a new graph polynomial in two variables. This "interlace" 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 significant differences are that our expansion is over vertex rather than edge subsets, and the rank and nullity employed are those of an adjacency matrix rather than an incidence matrix.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 interlace polynomial", is the single-variable interlace graph polynomial we studied previously, closely related to the Tutte-Martin polynomial on isotropic systems previously considered by Bouchet. Another, the "vertex-rank interlace polynomial", is equally interesting. Yet another specialization of the two-variable polynomial is the independent-set polynomial.

One and Two-Variable Interlace Polynomials: A Spectral Interpretation

“…As can be deduced from [4], the degree of q(1, y) is equal to max|IS|. The lemma follows by taking into account lemma 11.…”

“…We offer a definition of q(x, y) equivalent to the one proposed in [4]: Definition 15 The 2-variable interlace polynomial q(G; x, y) of a graph G in n variables is defined as…”

“…In [4], Arratia, Bollobas and Sorkin defined an extension of the interlace polynomial q, defined by themselves in [3], to a new polynomial q(x, y). Here we propose a similar extension of Q as defined by Aigner and Van der Holst in [1] to a new polynomial Q(x, y).…”

Computing Graph Polynomials on Graphs of Bounded Clique-Width

“…Polynomial tree-width k clique-width k Note that in the case of the matching polynomial f is linear in k, whereas for the chromatic polynomial it is exponential in k. For arbitrary graphs only the characteristic polynomial is in P. The others are all P-hard, [Val79], [ABS04b], [JVW90].…”

“…[Bol99], [GR01] and [ABS04b]. All these polynomials are not only of graph theoretic interest, but all of them have been motivated by or found applications to problems in chemistry, physics and biology.…”