Deza graphs: a survey and new results

Abstract

In this paper we survey existing results on Deza graphs and give some new results. We present an introduction to Deza graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of Deza graphs since the initial paper by five authors [M. Erickson, S. Fernando, W. H. Haemers, D. Hardy, J. Hemmeter, Deza graphs: A generalization of strongly regular graphs, J. Comb. Designs. 7 (1999), 395-405.] was written. We then investigate 3-class cyclotomic schemes and give necessary and sufficient conditions to get a Deza graph as a graph given by one relation or the union of two relations. Finally, we prove that a strictly Deza circulant on 2p vertices, where p is prime, is isomorphic to the lexicographical product of the Paley graph on p vertices with an edge.

The transformation of edge-regular and pseudo strongly regular graphs under graph operations

“…Recent research has focussed on the clique structure of such graphs ( [11,21]). A graph such that the number of common neighbours of any pair of vertices can be one of just two values is called a Deza graph with parameters (n, λ, µ); this family has attracted considerable recent interest ( [7,8,10]). A graph satisfying the conditions (1) and (3) has been mentioned in [18], but to our knowledge has not been given any specific name.…”

On WL-rank and WL-dimension of some Deza dihedrants

“…Deza graphs have been actively studied during the last years. For more details on Deza graphs and the recent progress in their studying, we refer the readers to the survey paper [16].…”