Genus Polynomials of Cubic Graphs with Non-real Roots

Abstract

Given a graph G, its genus polynomial is \(\Gamma _G(x) = \sum _{k\ge 0} g_k(G)x^k\), where \(g_k(G)\) is the number of two-cell embeddings of G in an orientable surface of genus k. The Log-Concavity Genus Distribution (LCGD) Conjecture states that the genus polynomial of every graph is log-concave. It was further conjectured by Stahl that the genus polynomial of every graph has only real roots, however, this was later disproved. We identify several examples of cubic graphs whose genus polynomials, in addition to having at least one non-real root, have a quadratic factor that is non-log-concave when factored over the real numbers.