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(r−1),1;1,μ,k}, and
any two adjacent vertices of Γ have exactly λ=k−1−μ(r−1) 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 g∈G−H are given. Let Γ(G,H,HgH) denote
the graph with vertex set {Hx|x∈G} whose edges are the pairs {Hx,Hy}
such that xy−1∈HgH.
The following theorem holds.
Theorem.
Let G=Sz(q), where q=22a+1>2, let S∈Syl2(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,q2−q−2,1;1,q+1,q2}.
Acknowledgement. The research was supported
by a grant from the IMM of UB RAS for young scientists in 2013.
References
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)
A. E. Brouwer, A. M. Cohen, A. Neumaier, Distance-regular graphs, Springer-Verlag, Berlin, (1989),
495 p.