M-Polynomial and Degree-Based Topological Indices

TL;DR

This paper introduces the M-polynomial, a new graph polynomial that simplifies the calculation of degree-based topological indices, demonstrated through examples.

Contribution

The paper presents the M-polynomial as a novel tool for efficiently computing degree-based topological indices from a single polynomial representation.

Findings

M-polynomial can be used to compute various topological indices.

The approach simplifies calculations compared to traditional methods.

Examples illustrate the effectiveness of the new method.

Abstract

Let $G$ be a graph and let $m_{ij}(G)$, $i,j\ge 1$, be the number of edges $uv$ of $G$ such that $\{d_v(G), d_u(G)\} = \{i,j\}$. The {\em $M$-polynomial} of $G$ is introduced with $\displaystyle{M(G;x,y) = \sum_{i\le j} m_{ij}(G)x^iy^j}$. It is shown that degree-based topological indices can be routinely computed from the polynomial, thus reducing the problem of their determination in each particular case to the single problem of determining the $M$-polynomial. The new approach is also illustrated with examples.