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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2012, Volume 18, Number 3, Pages 164–178
(Mi timm850)
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This article is cited in 2 scientific papers (total in 2 papers)
On distance-regular graphs on the set of nontrivial $p$-elements of the group $L_2(p^n)$
I. T. Mukhamet'yanov Lys'va Branch, Perm State National Research Polytechnical University
Abstract:
Let $\mathbf\Gamma_B$ be the graph with vertex set $B=g^G\cup(g^{-1})^G$, where $g^G$ is the class of conjugate elements of order $p$ of the group $G=L_2(p^n)$, and edge set $\{\{x,y\}\mid xy^{-1}\in B\}$; here, $p$ is an odd prime such that $p^n\geq5$. This graph was studied in some of the author's papers.
In this paper we clarify the structure of the graph $\mathbf\Gamma_B$ and describe the graph $\mathbf\Gamma_J$ whose vertex set is the set of elements of order $p$ of the group $G$ and edge set is $\{\{x,y\}\mid xy^{-1}\in J\}$, where $J$ is the class of adjoint involutions of $G$. In particular, we show that, in some cases, this graph is the union of two (isomorphic to each other) distance-regular graphs and, in other cases, its graph of $2$-distances is strongly regular.
Keywords:
graph, strongly regular graph, distance-regular graph, group.
Received: 17.05.2011
Citation:
I. T. Mukhamet'yanov, “On distance-regular graphs on the set of nontrivial $p$-elements of the group $L_2(p^n)$”, Trudy Inst. Mat. i Mekh. UrO RAN, 18, no. 3, 2012, 164–178
Linking options:
https://www.mathnet.ru/eng/timm850 https://www.mathnet.ru/eng/timm/v18/i3/p164
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