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 Trudy Inst. Mat. i Mekh. UrO RAN, 2010, Volume 16, Number 3, Pages 117–120 (Mi timm581)

On Deza graphs with parameters of lattice graphs

V. V. Kabanova, L. V. Shalaginovb

a Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences
b Chelyabinsk State University

Abstract: A Deza graph with parameters $(v,k,b,a)$, where $b\ge a$, is a $k$-regular graph on $v$ vertices in which any two vertices have either $a$ or $b$ common neighbors. A strongly regular graph with parameters $(v,k,\lambda,\mu)$ is a $k$-regular graph on $v$ vertices in which any two adjacent vertices have exactly $\lambda$ common neighbors and any two nonadjacent vertices have $\mu$ common neighbors. An strictly Deza graph is a Deza graph of diameter 2 that is not strongly regular. If a strongly regular graph has an involutive automorphism that transposes nonadjacent vertices only, then it is known that this automorphism can be used to obtain a Deza graph with the parameters of the initial strongly regular graph. We find all the automorphisms of strongly regular lattice $n\times n$ graphs with $n\ge3$ that satisfy the above condition. It turns out that there is exactly one such automorphism for odd $n$ and exactly two automorphisms for even $n$. Neighborhoods of exact Deza graphs obtained by means of this automorphism are found and a characterization of such strictly Deza graph with respect to its parameters and the structure of neighborhoods is obtained.

Keywords: line graph, strongly regular graph, Deza graph, strictly Deza graph, involutive automorphism.

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Document Type: Article
UDC: 519.174

Citation: V. V. Kabanov, L. V. Shalaginov, “On Deza graphs with parameters of lattice graphs”, Trudy Inst. Mat. i Mekh. UrO RAN, 16, no. 3, 2010, 117–120

Citation in format AMSBIB
\Bibitem{KabSha10}
\by V.~V.~Kabanov, L.~V.~Shalaginov
\paper On Deza graphs with parameters of lattice graphs
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2010
\vol 16
\issue 3
\pages 117--120
\mathnet{http://mi.mathnet.ru/timm581}
\elib{http://elibrary.ru/item.asp?id=15173469}

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. S. V. Goryainov, L. V. Shalaginov, “On Deza graphs with parameters of complement graphs to lattice and triangular graphs”, J. Appl. Industr. Math., 7:3 (2013), 355–362
2. A. L. Gavrilyuk, S. V. Goryainov, V. V. Kabanov, “On the vertex connectivity of Deza graphs”, Proc. Steklov Inst. Math. (Suppl.), 285, suppl. 1 (2014), S68–S77
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