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M-Polynomial of Product Graphs

El-Mehdi Mehiri, Sandi Klavžar

Abstract

The M-polynomial provides a unifying framework for a wide class of degree-based topological indices. Despite its structural importance, general methods for computing the M-polynomial under graph constructions remain limited. In this paper, explicit formulas, and compact ones whenever possible, for the M-polynomial under different graph products whose vertex sets are the Cartesian product of the factors are developed. The products studied are the direct, the Cartesian, the strong, the lexicographic, the symmetric-difference, the disjunction, and the Sierpiński product. The obtained formulas yield a unified structural description of how vertex-degree interactions propagate under graph constructions and extend existing results for degree-based indices at the polynomial level.

M-Polynomial of Product Graphs

Abstract

The M-polynomial provides a unifying framework for a wide class of degree-based topological indices. Despite its structural importance, general methods for computing the M-polynomial under graph constructions remain limited. In this paper, explicit formulas, and compact ones whenever possible, for the M-polynomial under different graph products whose vertex sets are the Cartesian product of the factors are developed. The products studied are the direct, the Cartesian, the strong, the lexicographic, the symmetric-difference, the disjunction, and the Sierpiński product. The obtained formulas yield a unified structural description of how vertex-degree interactions propagate under graph constructions and extend existing results for degree-based indices at the polynomial level.
Paper Structure (13 sections, 10 theorems, 55 equations, 1 figure)

This paper contains 13 sections, 10 theorems, 55 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph Products
  4. Formulas
  5. Cartesian Product
  6. Direct Product
  7. Strong Product
  8. Lexicographic Product
  9. Symmetric-Difference Product
  10. Disjunction Product
  11. Sierpiński Product
  12. Products of Paths
  13. Conclusion and Perspectives

Key Result

Theorem 3.1

If $G$ and $H$ are graphs, then

Figures (1)

  • Figure 1: The graph products considered for the case $G\cong H \cong P_3$. In the Sierpiński product $G\otimes_f H$, the function $f$ is defined by $f(1)=a$, $f(2)=b$, and $f(3)=c$.

Theorems & Definitions (20)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • proof
  • Theorem 3.5
  • proof
  • ...and 10 more