Deza graphs with parameters (n,k,k-1,a) and β=1
Goryainov,Sergey Haemers,Willem H. Kabanov,Vladislav V. Shalaginov,Leonid
Goryainov,Sergey
Haemers,Willem H.
Kabanov,Vladislav V.
Shalaginov,Leonid
Abstract
A Deza graph with parameters (n, k, b, a) is a k-regular graph with n vertices, in which any two vertices have a or b (a 1, where beta is the number of vertices with b common neighbours with a given vertex. Here, we start with a characterisation of Deza graphs (not necessarily strictly Deza graphs) with parameters (n, k, k - 1, 0). Then, we deal with the case beta = 1 and a > 0, and thus complete the characterisation of Deza graphs with b = k - 1. It follows that all Deza graphs with b = k - 1, beta = 1 and a > 0 can be made from special strongly regular graphs, and in fact are strictly Deza except for K-2. We present several examples of such strongly regular graphs. A divisible design graph (DDG) is a special Deza graph, and a Deza graph with beta = 1 is a DDG. The present characterisation reveals an error in a paper on DDGs by the second author et al. We discuss the cause and the consequences of this mistake and give the required errata.
Description
Date
2019-03
Journal Title
Journal ISSN
Volume Title
Publisher
Research Projects
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Journal Issue
Keywords
Deza graph, divisible design graph, dual Seidel switching, involution, strongly regular graph
Citation
Goryainov, S, Haemers, W H, Kabanov, V V & Shalaginov, L 2019, 'Deza graphs with parameters (n,k,k-1,a) and β=1', Journal of Combinatorial Designs, vol. 27, no. 3, pp. 188-202. https://doi.org/10.1002/jcd.21644