轮图亏格分布的渐近正态性与单峰性
On the Asymptotic Normality and Unimodality of Genus Distributions of Wheels
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中文总结 AI 辅助
本文结合联合树模型与特征理论推导轮图亏格多项式的显式公式并证明其实根性，证实了轮图亏格分布的单峰性猜想，还肯定回答了Zhang等人提出的渐近正态性问题。
中文摘要 AI 辅助
图的亏格多项式是该图在每个可定向曲面上的不等价嵌入数的生成多项式。本文针对轮图的亏格多项式研究三个问题：亏格多项式的计算、其系数的单峰性与渐近正态性。我们结合联合树模型（joint tree model）与特征理论（characters theory）推导轮图亏格多项式的显式公式，随后证明其具有实根性。这一更强的结果蕴含其系数的对数凹性、单峰性与渐近正态性。由此，我们证实了轮图亏格分布的单峰性猜想，并对Zhang、Peng和Chen（《应用数学进展》（Adv. in Appl. Math.）第127卷，2021年，第102175页）提出的渐近正态性问题给出了肯定回答。
英文摘要
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).
