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.