# Which Haar graphs are Cayley graphs? (Seminar DM)

Which Haar graphs are Cayley graphs?

István Estélyi

Torek, 24. marca 2015, od 10h do 12h, Plemljev seminar, Jadranska 19

Povzetek: For a finite group G and a subset S of G, a dipole with |S| parallel arcs labeled with elements of S, considered as a voltage graph, admits a regular covering graph, denoted by H(G,S), which is a bipartite regular graph, called a Haar graph. If G is an abelian group, then H(G,S) is well-known to be a Cayley graph; however, there are examples of non-abelian groups G and subsets S when this is not the case.

In this talk I am going to address the problem of classifying finite non-abelian groups G with the property that every Haar graph H(G,S) is a Cayley graph. We will deduce an equivalent condition for H(G,S) to be a Cayley graph of a group containing G in terms of G, S and Aut(G). We will see that the dihedral groups, which are solutions to the above problem, are $Z_2^2$,D6,D8 and D10.