The isomorphism of generalized Cayley graphs on finite non-abelian simple groups

The isomorphism problem is a fundamental problem for algebraic and combinatorial structures, particularly in relation to Cayley graphs. Let Xi = GC(G, Si, alpha i), (i = 1, 2) be generalized Cayley graphs. If whenever X1 similar to= X2, it implies that alpha 2 = alpha 1 gamma and S2 = g-1S gamma 1 g alpha 2 for some g is an element of G and gamma is an element of Aut(G), then G is a strongly generalized Cayley isomorphism (GCI)-group. In this study, we defined (strongly, restricted) m-GCI-groups. These definitions are similar to those of m-CI-groups for Cayley graphs. Our main results demonstrate that a finite non-abelian simple group G is a restricted 2-GCI-group if and only if G is one of A5, L2(8), M11, Sz(8), or M23, and G is a 2-GCI-group if and only if G is A5 or L2(8).