A characterization of weight-regular partitions of graphs

A partition P={V1,…,Vm} of the vertex set V of a graph is regular if, for all i, j, the number of neighbors which a vertex in Vi has in the set Vj is independent of the choice of vertex in Vi. The natural generalization of a regular partition, which makes sense also for non-regular graphs, is the so-called weight-regular partition, which gives to each vertex u?V a weight which equals the corresponding entry ?u of the Perron eigenvector ?. In this work we investigate when a weight-regular partition of a graph is regular in terms of double stochastic matrices. Inspired by a characterization of regular graphs by Hoffman, we provide a new characterization of weight-regular partitions by using a Hoffman-like polynomial.