A classification of nonabelian simple 3-BCI-groups

For a finite group G and a subset S ⊆ G (possibly, S contains the identity of G), the bi-Cayley graph BCay(G, S) of G with respect to S is the graph with vertex set G × {0, 1} and with edge set {(x, 0), (sx, 1)|x ∈ G, s ∈ S}. A bi-Cayley graph BCay(G, S) is called a BCI-graph if, for any bi-Cayley graph BCay(G, T ), whenever BCay(G, S) ∼= BCay(G, T ) we have T = gS , for some g ∈ G, α ∈ Aut(G). A group G is called an m-BCI-group, if all bi-Cayley graphs of G of valency at most m are BCI-graphs. In this paper, we prove that a finite nonabelian simple group is a 3-BCI-group if and only if it is A5. © 2009 Elsevier Ltd. All rights reserved.