The M-polynomial, introduced by Deutsch and Klavžar in 2015, provides a unifying algebraic framework for the computation of numerous degree-based topological indices such as the Zagreb, Randic, harmonic, and forgotten indices. Despite its broad applications in chemical graph theory and network analysis, closed expressions of the M-polynomial remain unknown for many important graph families. In this work we derive, for the first time, a complete explicit expression of the M-polynomial of the generalized Hanoi graphs $H_p^n$ for arbitrary positive $p$ and $n$. Our derivation relies on a detailed combinatorial analysis of the occupancy-based structure of $H_p^n$, refined using Stirling and $2$-associated Stirling numbers to enumerate all configurations with prescribed singleton and multiton counts. We obtain closed formulas for all diagonal and off-diagonal coefficients of the M-polynomial and show how these expressions yield exact values of the main degree-based topological indices. The correctness of the formulas is supported through numerical computation in small instances. These results provide a complete degree-based description of $H_p^n$ and make their structural complexity fully accessible through the M-polynomial framework.

翻译：M-多项式由Deutsch和Klavžar于2015年提出，为计算众多基于度的拓扑指数（如Zagreb指数、Randic指数、调和指数与遗忘指数）提供了统一的代数框架。尽管其在化学图论与网络分析中应用广泛，但许多重要图族的M-多项式闭合表达式仍属未知。本文首次针对任意正整数p和n，推导出广义河内图$H_p^n$的完整显式M-多项式表达式。我们的推导基于对$H_p^n$占用结构的详细组合分析，并利用斯特林数与2-关联斯特林数精确定义具有指定单元素与多元素计数的所有构型。我们获得了M-多项式所有对角与非对角系数的闭合公式，并展示了这些表达式如何导出主要基于度的拓扑指数的精确值。通过小规模实例的数值计算验证了公式的正确性。这些结果提供了$H_p^n$基于度的完整描述，使其结构复杂性可通过M-多项式框架完全解析。