Building Graphs Whose Independence Polynomials Have Only Real Roots

Abstract

A stable set in a graph G is a set of pairwise non-adjacent vertices, and the stability number α(G) is the maximum size of a stable set in G. The independence polynomial of G is $$I(G; x) = s_{0}+s_{1}x+s_{2}x^{2}+\cdots+s_{\alpha}x^{\alpha},\alpha=\alpha(G),$$of graphs, a graph U is induced-universal for $${\mathcal{F}}$$if every graph in $${\mathcal{F}}$$is an induced subgraph of U. We give a construction for an induced-universal graph for the family of graphs on n vertices with degree at most r, which has $$Cn^{\lfloor (r+1)/2\rfloor}$$vertices and $$Dn^{2\lfloor (r+1)/2\rfloor -1}$$edges, where C and D are constants depending only on r. This construction is nearly optimal when r is even in that such an induced-universal graph must have at least cnr/2 vertices for some c depending only on r. Our construction is explicit in that no probabilistic tools are needed to show that the graph exists or that a given graph is induced-universal. The construction also extends to multigraphs and directed graphs with bounded degree.