Explicit M-Polynomial and Degree-Based Topological Indices of Generalized Hanoi Graphs

TL;DR

This paper derives explicit formulas for the M-polynomial of generalized Hanoi graphs, enabling precise calculation of various degree-based topological indices crucial in chemical graph theory and network analysis.

Contribution

First explicit derivation of the M-polynomial for generalized Hanoi graphs using combinatorial analysis and Stirling numbers, facilitating exact topological index computation.

Findings

Closed formulas for all M-polynomial coefficients of H_p^n.

Exact values of degree-based topological indices derived from formulas.

Numerical verification confirms correctness for small instances.

Abstract

The M-polynomial, introduced by Deutsch and Klav\v{z}ar 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.