Mathematical Research Seminar - Archive
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A Cayley graph Cay(G,S) is called a CI-graph if for every subset T
of G, if Cay(G,T) and Cay(G,S) are isomorphic, then T=f(S) for some automorphism f of G. The group G is called a DCI-group if every Cayley graph of G is a CI-graph, and it is called a CI-group if every undirected Cayley graph of G is a CI-graph. Although there is a restrictive list of potentional CI-groups (Li-Lu-Pálfy, 2007), only a few classes of groups have been proved to be indeed CI; in several cases the proof was obtained by studying the Schur rings over the given group. In my talk I will review the Schur ring method.
Generalized Cayley graphs were defined by D.Marušič, R. Scapellato and N. Zagaglia Salvi in 1992. They studied properties of such graphs, mostly related to double coverings of graph. They also posed a question whether there exists a generalized Cayley graph which is vertex-transitive but not Cayley graph.
