A characterization of weight-regular partitions of graphs

Abstract

A partition $P=\{V_1,\dots ,V_m\}$ of the vertex set V of a graph is regular if, for all i, j, the number of neighbors which a vertex in V_i has in the set V_j is independent of the choice of vertex in V_i. 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 $u\in V$ a weight which equals the corresponding entry ${\nu }_u$ of the Perron eigenvector ν. 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.