On the existence of arc-transitive distance-regular covers of cliques
with $\lambda=\mu$ related to Suzuki groups

L. Tsiovkina

Let $\Gamma$ be a distance-regular graph of diameter $3$ and let $\mu$ denote the number of common neighbours for any two vertices of $\Gamma$ at distance $2$. Recall that if $\Gamma$ is antipodal then $\Gamma$ is an $r$-fold cover of $(k+1)$-clique, $\Gamma$ has intersection array $\{k,\mu(r-1),1;1,\mu,k\}$, and any two adjacent vertices of $\Gamma$ have exactly $\lambda=k-1-\mu(r-1)$ common neighbours, where the parameter $k$ is the degree of graph $\Gamma$, see [2].

A graph is said to be arc-transitive (or edge-symmetric) if its automorphism group acts transitively on ordered pairs of adjacent vertices.

Arc-transitive antipodal distance-regular graphs of diameter $3$ with $\lambda=\mu$ were described in [1]. In particular, in [1] there were found three new potential series of graphs corresponding to groups $Sz(q), U_3(q), ^2G_2(q)$.

In the present work, we prove the existence of infinite series of distance-regular graphs related to groups $Sz(q)$, where $q=2^{2a+1}>2$, which appears to be new.

Suppose that a non-normal subgroup $H$ of a group $G$ and an element $g\in G-H$ are given. Let $\Gamma(G,H,HgH)$ denote the graph with vertex set $\{Hx\ |\ x\in G\}$ whose edges are the pairs $\{Hx,Hy\}$ such that $xy^{-1}\in HgH$.

The following theorem holds.

Theorem. Let $G=Sz(q)$, where $q=2^{2a+1}>2$, let $S\in Syl_2(G)$ and let $g$ be an involution of $G$ not contained in $S$. Then $\Gamma(G,S,SgS)$ is an arc-transitive antipodal distance-regular graph with intersection array $\{q^2,q^2-q-2,1;1,q+1,q^2\}$.

Acknowledgement. The research was supported by a grant from the IMM of UB RAS for young scientists in 2013.

References

1. A. A. Makhnev, D. V. Paduchikh, L. Yu. Tsiovkina, Edge-symmetric distance-regular covers of cliques with $\lambda=\mu$, Dokl. Akad. Nauk, 448, N1 (2013), 22–26. (Russian)
2. A. E. Brouwer, A. M. Cohen, A. Neumaier, Distance-regular graphs, Springer-Verlag, Berlin, (1989), 495 p.

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