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On the existence of arc-transitive distance-regular covers of cliques
with λ=μ related to Suzuki groups

L. Tsiovkina

Let Γ be a distance-regular graph of diameter 3 and let μ denote the number of common neighbours for any two vertices of Γ at distance 2. Recall that if Γ is antipodal then Γ is an r-fold cover of (k+1)-clique, Γ has intersection array {k,μ(r1),1;1,μ,k}, and any two adjacent vertices of Γ have exactly λ=k1μ(r1) common neighbours, where the parameter k is the degree of graph Γ, 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 λ=μ were described in [1]. In particular, in [1] there were found three new potential series of graphs corresponding to groups Sz(q),U3(q),2G2(q).

In the present work, we prove the existence of infinite series of distance-regular graphs related to groups Sz(q), where q=22a+1>2, which appears to be new.

Suppose that a non-normal subgroup H of a group G and an element gGH are given. Let Γ(G,H,HgH) denote the graph with vertex set {Hx | xG} whose edges are the pairs {Hx,Hy} such that xy1HgH.

The following theorem holds.

Theorem. Let G=Sz(q), where q=22a+1>2, let SSyl2(G) and let g be an involution of G not contained in S. Then Γ(G,S,SgS) is an arc-transitive antipodal distance-regular graph with intersection array {q2,q2q2,1;1,q+1,q2}.

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 λ=μ, 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.

See also the author's pdf version: pdf

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