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M-polynomial Based Mathematical Formulation of the Hyperbolic Sombor Index

Jayjit Barman, Shibsankar Das

TL;DR

This work derives a closed derivation formula that expresses the hyperbolic Sombor index $HSO(G)$ entirely in terms of a graph's M-polynomial, via the operation $HSO(G)=D^{1/2}_{x}\,JP_{y}P_{x}S_{x}(M(G;x,y))|_{x=1}$, enabling efficient computation for large graphs. The authors apply the formula to a broad spectrum of graphs and chemical families, providing explicit $M$-polynomials and closed-form $HSO$ expressions for each case (e.g., $K_{m,n}$, $K_{r,r}$, $C_n$, $P_n$, boron sheets, dendrimers, benzenoid systems, PAHs, V-Phynelenic structures, porous graphene, tadpoles, and polyphenylenes). They also present numerical and graphical representations of the M-polynomials and corresponding $HSO$ values, including MATLAB visualizations. The results enhance the toolkit for topology-based molecular modeling, enabling rapid QSPR/QSAR analyses by relating structural degrees to the hyperbolic Sombor index through the M-polynomial framework. Overall, the paper makes a practical contribution by providing ready-to-use closed forms for $HSO$ across diverse graph families, supporting topology-informed chemical informatics and material science applications.

Abstract

The numerical values extracted from a graph that indicates its topology are called topological indices. A contemporary and efficient method is to compute a graph's topological indices using the graph polynomial that corresponds to it. This method of identifying degree-based topological indices involves the use of the M-polynomial. Very recently, in 2025, the hyperbolic Sombor index (HSO) was proposed and shows its chemical applicability for octane isomers and the structure sensitivity and abruptness for octane, nonane, and decane isomers, respectively. In this work, we establish the closed derivation formula for the above-mentioned index of a graph based on its M-polynomial. Additionally, we use our proposed derivation formula to calculate the hyperbolic Sombor index of a few standard graphs and chemical families. Moreover, we provide the numerical and graphical representations for the M-polynomial and the computed HSO index of the chemical families.

M-polynomial Based Mathematical Formulation of the Hyperbolic Sombor Index

TL;DR

This work derives a closed derivation formula that expresses the hyperbolic Sombor index entirely in terms of a graph's M-polynomial, via the operation , enabling efficient computation for large graphs. The authors apply the formula to a broad spectrum of graphs and chemical families, providing explicit -polynomials and closed-form expressions for each case (e.g., , , , , boron sheets, dendrimers, benzenoid systems, PAHs, V-Phynelenic structures, porous graphene, tadpoles, and polyphenylenes). They also present numerical and graphical representations of the M-polynomials and corresponding values, including MATLAB visualizations. The results enhance the toolkit for topology-based molecular modeling, enabling rapid QSPR/QSAR analyses by relating structural degrees to the hyperbolic Sombor index through the M-polynomial framework. Overall, the paper makes a practical contribution by providing ready-to-use closed forms for across diverse graph families, supporting topology-informed chemical informatics and material science applications.

Abstract

The numerical values extracted from a graph that indicates its topology are called topological indices. A contemporary and efficient method is to compute a graph's topological indices using the graph polynomial that corresponds to it. This method of identifying degree-based topological indices involves the use of the M-polynomial. Very recently, in 2025, the hyperbolic Sombor index (HSO) was proposed and shows its chemical applicability for octane isomers and the structure sensitivity and abruptness for octane, nonane, and decane isomers, respectively. In this work, we establish the closed derivation formula for the above-mentioned index of a graph based on its M-polynomial. Additionally, we use our proposed derivation formula to calculate the hyperbolic Sombor index of a few standard graphs and chemical families. Moreover, we provide the numerical and graphical representations for the M-polynomial and the computed HSO index of the chemical families.
Paper Structure (13 sections, 37 theorems, 66 equations, 12 figures, 3 tables)

This paper contains 13 sections, 37 theorems, 66 equations, 12 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Closed Derivation Formula over the M-polynomial for the Hyperbolic Sombor Index
  3. Hyperbolic Sombor Index for Some Well-known Graphs and Chemical Families
  4. Hyperbolic Sombor Index for Some Well-known Graphs
  5. Hyperbolic Sombor Index for Boron Icosahedral $\boldsymbol{\alpha}$ sheet $\boldsymbol{\mathcal{B}_{\alpha}(a,b)}$
  6. Hyperbolic Sombor Index for Dendrimers
  7. Hyperbolic Sombor Index for Jagged-rectangle Benzenoid System $\boldsymbol{B_{m,n}}$
  8. Hyperbolic Sombor Index for Polycyclic Aromatic Hydrocarbon $\textit{PAH}_n$
  9. Hyperbolic Sombor Index for V-Phynelenic Nanotubes $\boldsymbol{\textit{VPHX}[m,n]}$ and V-Phynelenic Nanotori $\boldsymbol{\textit{VPHY}[m,n]}$
  10. Hyperbolic Sombor Index for Porous Graphene $\boldsymbol{\textit{PG}[p,q]}$
  11. Hyperbolic Sombor Index for Tadpole Graph $\boldsymbol{\textit{T}(n,m)}$
  12. Hyperbolic Sombor Index for Polyphenylenes $\boldsymbol{\textit{P}[s,t]}$
  13. Conclusion

Key Result

Theorem 1

Let $G=(V(G),E(G))$ be a graph and its hyperbolic Sombor index is then where $M(G(x,y))$ is the M-polynomial of $G$.

Figures (12)

  • Figure 1: Graphical illustration of the (a) M-polynomial of $\mathcal{B}_{\alpha}(5,5)$ and (b) HSO index of $\mathcal{B}_{\alpha}(a,b)$.
  • Figure 2: Graphical illustration of the (a) M-polynomial and (b) HSO index of PETIM dendrimer for $n=5$.
  • Figure 3: Graphical illustration of the (a) M-polynomial and (b) HSO index of $D_nP_n$ dendrimer for $n=5$.
  • Figure 4: Graphical illustration of the (a) M-polynomial and (b) HSO index of $\textit{DPZ}_n$ dendrimer for $n=5$.
  • Figure 5: Graphical illustration of the (a) M-polynomial and (b) HSO index of PETAA dendrimer for $n=5$.
  • ...and 7 more figures

Theorems & Definitions (55)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • proof
  • Corollary 4
  • Corollary 5
  • Theorem 6
  • Theorem 7
  • ...and 45 more