University of KashanIranian Journal of Mathematical Chemistry2228-648911220200730On the M-polynomial of planar chemical graphs657110605710.22052/ijmc.2020.224280.1492ENEmericDeutschPolytechnic Institute of New York University (retired)SandiKlavžarFaculty of Mathematics and Physics, University of Ljubljana, SloveniaJournal Article20200322Let $G$ be a graph and let $m_{i,j}(G)$, $i,jge 1$, be the number of edges $uv$ of $G$ such that ${d_v(G), d_u(G)} = {i,j}$. The $M$-polynomial of $G$ is $M(G;x,y) = sum_{ile j} m_{i,j}(G)x^iy^j$. With $M(G;x,y)$ in hands, numerous degree-based topological indices of $G$ can be routinely computed. In this note a formula for the $M$-polynomial of planar (chemical) graphs which have only vertices of degrees $2$ and $3$ is given that involves only invariants related to the degree $2$ vertices and the number of faces. The approach is applied on several families of chemical graphs. In one of these families an error from the literature is corrected.https://ijmc.kashanu.ac.ir/article_106057_81375683c320f653c28092d0868d56fb.pdf