On the Extendability of Quasi-Strongly Regular Graphs with Diameter 2

Abstract

Let $\ell$ denote a non-negative integer and let $\Gamma$ be a connected graph of even order at least $2 \ell +2$. It is said that $\Gamma$ is $\ell$-extendable if it contains a matching of size $\ell$ and if every such matching is contained in a perfect matching of $\Gamma$. A connected regular graph $\Gamma$ is quasi-strongly regular with parameters $(n, k, \lambda ; \mu _1, \mu _2, \ldots , \mu _s)$, if it is a k-regular graph on n vertices, such that any two adjacent vertices have exactly $\lambda$ common neighbours and any two distinct and non-adjacent vertices have exactly $\mu _i$ common neighbours for some $1 \le i \le s$. The grade of $\Gamma$ is the number of indices $1 \le i \le s$ for which there exist two distinct and non-adjacent vertices in $\Gamma$ with $\mu _i$ common neighbours. In this paper we study the extendability of quasi-strongly regular graphs of diameter 2 and grade 2. In particular, we classify the 2-extendable members of this class of graphs.