A bound on the values of independence polynomials at −1/k for k-degenerate graphs

An independent set of a graph G is a set of pairwise non-adjacent vertices. Let α(G) denote the cardinality of a maximum independent set and fs(G) for 0≤s≤α(G) denote the number of independent sets on s vertices. The independence polynomial I(G;x)=∑i=0α(G)fs(G)xs defined first by Gutman and Harary in 1983 has been the focus of considerable research. In 1995, Wingard bounded the function values obtained at −1 for the independence polynomials for the tree T; |I(T;−1)|≤1. We generalize Wingard's result for a much larger class of graphs, k-degenerate graphs, a class which includes all k-trees. Wingard's result is the case when k=1.