On the Asymptotic Normality and Unimodality of Genus Distributions of Wheels

Abstract

The genus polynomial of a graph is the generating polynomial for the number of nonequivalent embeddings of the graph on each orientable surface. In this paper, we address three questions on genus polynomials for wheel graphs: the computation of genus polynomials, the unimodality and the asymptotic normality of their coefficients. We derive an explicit formula for the genus polynomial of wheel graphs by combining methods of the joint tree model and characters theory, and then prove its real-rootedness. This stronger result implies the log-concavity, unimodality, and asymptotic normality of its coefficients. Thus, we confirm the unimodality conjecture for the genus distribution of wheel graphs and provide a positive answer to the asymptotic normality question posed by Zhang, Peng, and Chen (\emph{Adv. in Appl. Math.} \textbf{127} (2021), 102175).