Quasi-strongly regular graphs of grade three with diameter two

TL;DR

This paper investigates a specific class of quasi-strongly regular graphs of grade three with diameter two, establishing bounds on their order and characterizing graphs that meet these bounds.

Contribution

It provides sharp bounds on the order of quasi-strongly regular graphs of grade three with diameter two and characterizes all graphs that attain these bounds.

Findings

Established sharp bounds on the order n for given degree k.

Characterized all such graphs that meet the bounds.

Extended understanding of the structure of quasi-strongly regular graphs.

Abstract

A quasi-strongly regular graph of grade $p$ with parameters $(n, k, a; c_1, \ldots, c_p)$ is a $k$-regular graph of order $n$ such that any two adjacent vertices share $a$ common neighbours and any two non-adjacent vertices share $c_{i}$ common neighbours for some $1 \leq i \leq p$. This is a generalization of a strongly regular graph. In this paper, we focus on strictly quasi-strongly regular graphs of grade $3$ with $c_i = k - i$ for $i = 1, 2, 3$. The main result is to show the sharp bounds of order $n$ for a given $k \geq 4$. Furthermore, by this result, we characterize all of these graphs whose $n$ satisfies upper or lower bounds.