  In this paper we provide the group inverse of the combinatorial Laplacian
matrix of distance-biregular graphs using the so-called equilibrium measures
for sets obtained by deleting a vertex. We also show that the two equilibrium
arrays characterizing a distance-biregular graph can be expressed in terms of
the mentioned equilibrium measures. As a consequence of the minimum principle,
we show a characterization of when the group inverse of the combinatorial
Laplacian matrix of a distance-biregular graph is an $M$-matrix.
