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

El-Mehdi Mehiri

TL;DR

This work addresses the problem of obtaining a complete closed-form expression for the M-polynomial $M(H_p^n;x,y)$ of generalized Hanoi graphs $H_p^n$. It develops a occupancy-based combinatorial framework that counts vertex configurations and edge moves using Stirling numbers of the second kind and falling factorials, enabling explicit formulas for all diagonal and adjacent off-diagonal coefficients of $M(H_p^n;x,y)$. The authors derive a fully explicit expression for the M-polynomial and demonstrate that a wide class of degree-based topological indices (e.g., Zagreb, Randic, harmonic, forgotten) can be obtained in closed form via differential operators. The results provide a complete degree-based description of $H_p^n$ and offer a scalable method for analyzing self-similar Hanoi graphs and related graph families.

Abstract

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.

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

TL;DR

This work addresses the problem of obtaining a complete closed-form expression for the M-polynomial of generalized Hanoi graphs . It develops a occupancy-based combinatorial framework that counts vertex configurations and edge moves using Stirling numbers of the second kind and falling factorials, enabling explicit formulas for all diagonal and adjacent off-diagonal coefficients of . The authors derive a fully explicit expression for the M-polynomial and demonstrate that a wide class of degree-based topological indices (e.g., Zagreb, Randic, harmonic, forgotten) can be obtained in closed form via differential operators. The results provide a complete degree-based description of and offer a scalable method for analyzing self-similar Hanoi graphs and related graph families.

Abstract

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 for arbitrary positive and . Our derivation relies on a detailed combinatorial analysis of the occupancy-based structure of , refined using Stirling and -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 and make their structural complexity fully accessible through the M-polynomial framework.

Paper Structure

This paper contains 12 sections, 31 theorems, 95 equations, 1 figure, 9 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalized Hanoi graphs
  4. Degree-based combinatorial analysis
  5. Dependence of vertex degree on occupancy
  6. Adjacency relations between degree classes
  7. Enumeration of configurations by occupancy
  8. Edge decomposition and degree-based classification
  9. Explicit expression of the M-polynomial
  10. Example 1: $H_3^3$.
  11. Example 2: $H_4^2$.
  12. Conclusion

Key Result

Lemma 1

For a vertex $s$ with occupancy $o(s)=\mu$, its degree in $H_p^n$ is given by

Figures (1)

  • Figure 1: Examples of Hanoi graphs $H_3^3$ and $H_4^2$. Each vertex represents a regular configuration of discs, and edges represent single legal moves.

Theorems & Definitions (70)

  • Definition 1: M-polynomial EmericMploy
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6: hinz2018tower
  • Definition 7: Occupancy and structural parameters
  • Lemma 1: Degree as a function of occupancy HINZ20121521Arett01062010
  • proof
  • Corollary 1: Degree bounds
  • ...and 60 more