Spectra of strongly Deza graphs

TL;DR

This paper characterizes the spectra of strongly Deza graphs, explores relationships between their eigenvalues, and investigates their properties when they are also distance-regular, advancing understanding of their algebraic structure.

Contribution

It provides a spectral characterization of strongly Deza graphs and examines their eigenvalues and distance-regularity properties, which are novel insights in algebraic graph theory.

Findings

Spectral characterization of strongly Deza graphs

Relationships between eigenvalues of Deza graphs

Analysis of strongly Deza graphs that are distance-regular

Abstract

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.