# 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

, Volume 34 (4) – May 30, 2018
16 pages

/lp/springer_journal/on-the-extendability-of-quasi-strongly-regular-graphs-with-diameter-2-b8UMnQ9BM3
Publisher
Springer Journals
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

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations