Deza graphs: A generalization of strongly regular graph

Abstract

Abstract:We consider the following generalization of strongly regular graphs. A graph G is a Deza graph if it is regular and the number of common neighbors of two distinct vertices takes on one of two values (not necessarily depending on the adjacency of the two vertices). We introduce several ways to construct Deza graphs, and develop some basic theory. We also list all diameter two Deza graphs which are not strongly regular and have at most 13 vertices.

On strictly Deza graphs with parameters (n,k,k−1,a)

“…In any case, k + 3 + 6 k − 3 > k + 4, then k < 9 and v < 17. But all such Deza graphs are known [5,7]. In this case G is one of the strictly Deza graphs with parameters (8, 4, 2, 1) or (9, 4, 2, 1).…”

“…The authors of [5] developed a basic theory of strictly Deza graphs and introduced a few constructions of such graphs. They also found all strictly Deza graphs with the number of vertices at most 13.…”

“…Strongly regular graphs with parameters (v, k, λ, µ) such that k = µ are known (Theorem 1.3.1(v) in [1]). Deza graphs with parameters (v, k, b, a) such that b = k was obtained in the paper [5,Theorem 2.6]. Deza graphs with b = k − 1 were characterised in [9] and [6].…”

Deza graphs with parameters and

“…for all quadruples x x x x { , ′, , ′} b b of NA-vertices. It follows from [4] that this permutation is an order 2 automorphism of Γ‴ that interchanges only nonadjacent vertices. Thus, Γ″ can be obtained from Γ‴ by dual Seidel switching, which implies that Γ comes from Construction 2 by definition.…”

“…Because of this, the discovered mistake in [7] has some consequences for Table 1 in [7] and [3], and the second author would like to take the opportunity to give the necessary corrections. For the parameter sets (18, 9,8,4) and (20, 7, 6, 2), there can occur an additional eigenvalue 1, and the multiplicities are not determined by the parameters. Moreover, for these parameter sets, the "no" occurring in the column headed by "notWR" in Table 1 of [3] should be replaced by "yes.…”