Genus Polynomials of Cubic Graphs with Non-Real Roots

TL;DR

This paper investigates the roots of genus polynomials of cubic graphs, providing examples with non-real roots and non-log-concave quadratic factors, challenging previous conjectures about their properties.

Contribution

It identifies specific cubic graphs with genus polynomials that have non-real roots and non-log-concave quadratic factors, advancing understanding of their algebraic structure.

Findings

Genus polynomials of certain cubic graphs have non-real roots.

Some genus polynomials contain quadratic factors that are not log-concave.

Counterexamples to previous conjectures about genus polynomial roots.

Abstract

Given a graph $G$, its genus polynomial is $\Gamma_G(x) = \sum_{k\geq 0} g_k(G)x^k$, where $g_k(G)$ is the number of 2-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.