M-Polynomial and Degree-Based Topological Indices

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 $M(G;x,y)=\sum _{i\le j}{m}_{ij}(G)x^i{y}^j$. 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.