Cayley graphs of finite groups

AbstractLet Γ(G, T) denote the Cayley graph of a finite group G with respect to a normal subset T of G−{1}.We compute explicitly the spectrum of Γ(G, T) in terms of complex character values. This allows us to determine the number of paths of length n between two arbitrary vertices of Γ(G, T) for each n ∈ N.Finally, we apply these results to obtain the following theorem. Suppose that G is a finite group which contains a cyclic self-normalizing subgroup W of order pq, where p and q are two different odd prime numbers. Define W0 to be the set of all elements of order pq of W and let T:=∪gϵG W0g. Then for any n ∈ N, the number of paths of length n between two adjacent vertices of the Cayley graph Γ(G, T) does not depend on the choice of the two adjacent vertices. Moreover, the rank of Γ(G, T) is 4 or 5