On isomorphisms of small order bi-Cayley graphs

Abstract

For a finite group G and a subset S⊇G (possibly, S contains the identity), 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^α, for some g ∈G and α ∈ Aut(G). A group G is called a BCI-group, if all bi-Cayley graphs of G are BCI-graphs. In this paper, we prove that except for the dihedral group D_s, the cyclic group Z_s and the abelian group Z₄ × Z₂, all groups of order less than 9 are BCI-groups. Since Z₈ is a CI-group but not a BCI-group, it also shows that a CI-group is not necessary to be a BCI-group.