  A natural generalization of a regular (or equitable) partition of a graph,
which makes sense also for non-regular graphs, is the so-called weight-regular
partition, which gives to each vertex $u\in V$ a weight that equals the
corresponding entry $\nu_u$ of the Perron eigenvector $\mathbf{\nu}$. This
paper contains three main results related to weight-regular partitions of a
graph. The first is a characterization of weight-regular partitions in terms of
double stochastic matrices. Inspired by a characterization of regular graphs by
Hoffman, we also provide a new characterization of weight-regularity by using a
Hoffman-like polynomial. As a corollary, we obtain Hoffman's result for regular
graphs. In addition, we show an application of weight-regular partitions to
study graphs that attain equality in the classical Hoffman's lower bound for
the chromatic number of a graph, and we show that weight-regularity provides a
condition under which Hoffman's bound can be improved.
