  The $M$-polynomial of a graph $G$ is defined as $\sum_{i\le j}
m_{i,j}(G)x^iy^j$, where $m_{i,j}(G)$, $i,j\ge 1$, is the number of edges $uv$
of $G$ such that $\{d_v(G), d_u(G)\} = \{i,j\}$. Knowing the $M$-polynomial,
formulas for bond incident degree indices (an important subclass of
degree-based topological indices) can be obtained by means of specific
operators defined on differentiable functions in two variables. This is
illustrated on three infinite families of Bethe cacti. Gutman's approach for
the computation of the coefficients of the $M$-polynomial is also recalled and
an extension of it is given. This extension is used to determine the
$M$-polynomial of a two-parameter infinite family of lattice graphs.
