On the M-polynomial of Planar Chemical Graphs

Document Type : Research Paper


1 Polytechnic Institute of New York University (retired)

2 Faculty of Mathematics and Physics, University of Ljubljana, Slovenia


Let $G$ be a graph and let $m_{i,j}(G)$, $i,j\ge 1$, be the number of edges $uv$ of $G$ such that $\{d_v(G), d_u(G)\} = \{i,j\}$. The $M$-polynomial of $G$ is $M(G;x,y) = \sum_{i\le j} m_{i,j}(G)x^iy^j$. 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.


Main Subjects

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