Spectra of strongly Deza graphs

A Deza graph G with parameters (n, k, b, a) is a k-regular graph with n vertices such that any two distinct vertices have b or a common neighbours. The children G(A) and G(B) of a Deza graph G are defined on the vertex set of G such that every two distinct vertices are adjacent in G(A) or G(B) if and only if they have a or b common neighbours, respectively. A strongly Deza graph is a Deza graph with strongly regular children. In this paper we give a spectral characterisation of strongly Deza graphs, show relationships between eigenvalues, and study strongly Deza graphs which are distance-regular. (C) 2021 Elsevier B.V. All rights reserved.