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
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)
A. E. Brouwer, A. M. Cohen, A. Neumaier, Distance-regular graphs, Springer-Verlag, Berlin, (1989),
495 p.
