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

On the Extendability of Quasi-Strongly Regular Graphs with Diameter 2 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Graphs and Combinatorics Springer Journals

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

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Japan KK, part of Springer Nature
Subject
Mathematics; Combinatorics; Engineering Design
ISSN
0911-0119
eISSN
1435-5914
D.O.I.
10.1007/s00373-018-1908-6
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

Graphs and CombinatoricsSpringer Journals

Published: May 30, 2018

References

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