How to Compute the M-Polynomial of (Chemical) Graphs

Let G be a graph and let m i,j ( G ), i, j ≥ 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 ) = (cid:80) i ≤ j m i,j ( G ) x i y j . A general method for calculating the M-polynomials for arbitrary graph families is presented. The method is further developed for the case where the vertices of a graph have degrees 2 and p , where p ≥ 3, and further for such planar graphs. The method is illustrated on families of chemical graphs.