On the \(p\)-ranks of the adjacency matrices of distance-regular graphs
type: Item
instance of: scholarly article
title: On the \(p\)-ranks of the adjacency matrices of distance-regular graphs
zbMATH Open document ID: 0989.05074
DOI: 10.1023/A:1013842904024
author: René Peeters
published in: Journal of Algebraic Combinatorics
publication date: 2002-05-30T00:00:00Z
review text: Let \(\Gamma\) be a distance-regular graph with adjacency matrix \(A\). Let \(I\) be the identity matrix, \(J\) the all-1 matrix, and let \(p\) be a prime. The author considers the \(p\)-rank of the matrices \(A+bJ-cI\), where \(b\) and \(c\) are integers, generalizing results by \textit{A. E. Brouwer} and \textit{C. A. van Eijl} [J. Algebr. Comb. 1, No. 4, 329--346 (1992; Zbl 0780.05039)]. Using the minimal polynomial of \(A\), as well as the theory of Smith normal forms it is shown that most of these ranks follow directly from the spectrum. Those ranks that, in general, do not follow from the spectrum only, depend also on the actual structure of the graph and can be used to distinguish between cospectral graphs. The author presents complete results for Hamming and Doob graphs, as well as further results for few other classes of distance-regular graphs, such as Johnson graphs, generalized quadrangles minus a spread, square \(2\)-designs and Taylor graphs.
reviewed by: Dragan Stevanović
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 05E30
Mathematics Subject Classification ID: 05C75
zbMATH DE Number: 1747858
zbMATH Keywords: \(p\)-rank
zbMATH Keywords: distance-regular graph
zbMATH Keywords: adjacency matrix
zbMATH Keywords: minimal polynomial
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1747858