On distance-regular graphs on the set of nontrivial $p$-elements of the group~$L_2(p^n)$

I. T. Mukhamet'yanov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 3, pp. 164-178

Résumé

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.

Détail

Keywords: graph, strongly regular graph, distance-regular graph
Mots-clés : group.