Linear Recurrence Relations for Graph Polynomials

A sequence of graphs Gn is iteratively constructible if it can be built from an initial labeled graph by means of a repeated fixed succession of elementary operations involving addition of vertices and edges, deletion of edges, and relabelings. Let Gn be a iteratively constructible sequence of graphs. In a recent paper, [27], M. Noy and A. Ribò have proven linear recurrences with polynomial coefficients for the Tutte polynomials T (Gi, x, y) = T (Gi), i.e. T (Gn+r) = p1(x, y)T (Gn+r−1) + . . .+ pr(x, y)T (Gn). We show that such linear recurrences hold much more generally for a wide class of graph polynomials (also of labeled or signed graphs), namely they hold for all the extended MSOL-definable graph polynomials. These include most graph and knot polynomials studied in the literature.