Let ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document} denote a non-negative integer and let Γ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document} be a connected graph of even order at least 2ℓ+2\documentclass[12pt]{minimal}
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\begin{document}$$2 \ell +2$$\end{document}. It is said that Γ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document} is ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document}-extendable if it contains a matching of size ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document} and if every such matching is contained in a perfect matching of Γ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document}. A connected regular graph Γ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document} is quasi-strongly regular with parameters (n,k,λ;μ1,μ2,…,μs)\documentclass[12pt]{minimal}
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\begin{document}$$(n, k, \lambda ; \mu _1, \mu _2, \ldots , \mu _s)$$\end{document}, if it is a k-regular graph on n vertices, such that any two adjacent vertices have exactly λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document} common neighbours and any two distinct and non-adjacent vertices have exactly μi\documentclass[12pt]{minimal}
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\begin{document}$$\mu _i$$\end{document} common neighbours for some 1≤i≤s\documentclass[12pt]{minimal}
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\begin{document}$$1 \le i \le s$$\end{document}. The grade of Γ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document} is the number of indices 1≤i≤s\documentclass[12pt]{minimal}
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\begin{document}$$1 \le i \le s$$\end{document} for which there exist two distinct and non-adjacent vertices in Γ\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document} with μi\documentclass[12pt]{minimal}
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\begin{document}$$\mu _i$$\end{document} 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.
