Let $$\ell $$ ℓ denote a non-negative integer and let $$\Gamma $$ Γ be a connected graph of even order at least $$2 \ell +2$$ 2 ℓ + 2 . It is said that $$\Gamma $$ Γ is $$\ell $$ ℓ -extendable if it contains a matching of size $$\ell $$ ℓ and if every such matching is contained in a perfect matching of $$\Gamma $$ Γ . A connected regular graph $$\Gamma $$ Γ is quasi-strongly regular with parameters $$(n, k, \lambda ; \mu _1, \mu _2, \ldots , \mu _s)$$ ( n , k , λ ; μ 1 , μ 2 , … , μ s ) , if it is a k-regular graph on n vertices, such that any two adjacent vertices have exactly $$\lambda $$ λ common neighbours and any two distinct and non-adjacent vertices have exactly $$\mu _i$$ μ i common neighbours for some $$1 \le i \le s$$ 1 ≤ i ≤ s . The grade of $$\Gamma $$ Γ is the number of indices $$1 \le i \le s$$ 1 ≤ i ≤ s for which there exist two distinct and non-adjacent vertices in $$\Gamma $$ Γ with $$\mu _i$$ μ 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.
Graphs and Combinatorics – Springer Journals
Published: May 30, 2018
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