Explicit M-Polynomial and Degree-Based Topological Indices of Generalized Hanoi Graphs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917365280669696 |
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| author | Mehiri, El-Mehdi |
| author_facet | Mehiri, El-Mehdi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_12587 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explicit M-Polynomial and Degree-Based Topological Indices of Generalized Hanoi Graphs Mehiri, El-Mehdi Combinatorics Discrete Mathematics 05C31, 05C76, 05A15, 05C07 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. |
| title | Explicit M-Polynomial and Degree-Based Topological Indices of Generalized Hanoi Graphs |
| topic | Combinatorics Discrete Mathematics 05C31, 05C76, 05A15, 05C07 |
| url | https://arxiv.org/abs/2511.12587 |