Abstract
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 \(M(G;x,y) = \sum_{i\le j} m_{i,j}(G)x^iy^j\). 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 \(p\ge 3\), and further for such planar graphs. The method is illustrated
on families of chemical graphs.