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

TL;DR

This paper investigates the properties and parameters of strictly Deza graphs with specific conditions, focusing on their structure when the parameters satisfy certain algebraic relations.

Contribution

It characterizes strictly Deza graphs with parameters where $k = b + 1$ and a particular inequality involving $k$, $a$, and $n$, expanding understanding of their structure.

Findings

Identifies conditions under which strictly Deza graphs exist.

Provides structural insights for graphs with given parameters.

Establishes new bounds and properties for these graphs.

Abstract

A nonempty $k$-regular graph $\Gamma$ on $n$ vertices is called a Deza graph if there exist constants $b$ and $a$ $(b \geq a)$ such that any pair of distinct vertices of $\Gamma$ has precisely either $b$ or $a$ common neighbours. The quantities $n$, $k$, $b$, and $a$ are called the parameters of $\Gamma$ and are written as the quadruple $(n,k,b,a)$. If a Deza graph has diameter 2 and is not strongly regular, then it is called a strictly Deza graph. In the paper we investigate strictly Deza graphs with parameters $ (n, k, b, a) $, where its quantities satisfy the conditions $k = b + 1$ and $\frac{k(k - 1) - a(n - 1)}{b - a} > 1$.