M-Polynomial Revisited: Bethe Cacti and an Extension of Gutman's Approach

TL;DR

This paper revisits the $M$-polynomial concept, extends Gutman's method for its calculation, and applies these to Bethe cacti and lattice graphs to derive formulas for degree-based topological indices.

Contribution

It introduces an extension of Gutman's approach for computing the $M$-polynomial and applies it to infinite families of graphs, including Bethe cacti and lattice graphs.

Findings

Derived $M$-polynomials for Bethe cacti.

Extended Gutman's method for broader graph classes.

Calculated degree-based indices from the $M$-polynomial.

Abstract

The $M$-polynomial of a graph $G$ is defined as $\sum_{i\le j} m_{i,j}(G)x^iy^j$, where $m_{i,j}(G)$, $i,j\ge 1$, is the number of edges $uv$ of $G$ such that $\{d_v(G), d_u(G)\} = \{i,j\}$. Knowing the $M$-polynomial, formulas for bond incident degree indices (an important subclass of degree-based topological indices) can be obtained by means of specific operators defined on differentiable functions in two variables. This is illustrated on three infinite families of Bethe cacti. Gutman's approach for the computation of the coefficients of the $M$-polynomial is also recalled and an extension of it is given. This extension is used to determine the $M$-polynomial of a two-parameter infinite family of lattice graphs.