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A characterization and an application of weight-regular partitions of graphs

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Abstract

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 is an element of V a weight that equals the corresponding entry nu(u) of the Perron eigenvector 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. (C) 2019 Elsevier Inc. All rights reserved.

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  • Weight-regular partition, Hoffman polynomial, Chromatic number, ALGEBRAIC CHARACTERIZATIONS
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Details

Original languageEnglish
Pages (from-to)162-174
Number of pages13
JournalLinear Algebra and Its Applications
Volume569
DOIs
Publication statusPublished - 15 May 2019