Cayley graphs of finite groups (Q1115972)

Cayley graphs \(\Gamma\) (G,T) of a finite group G defined by normal generating sets T (unions of conjugacy classes) are studied via irreducible complex characters. The author obtains a formula equating the eigenvalues of the Cayley graph to weighted sums of the character values at points over T. As a consequence, he provides a formula to compute the number of paths of length n connecting 1 to any vertex in G. Under the assumption that G possesses a cyclic subgroup of odd order, this number is an invariant for elements of the same order. The number is also invariant for adjacent elements of the Cayley graph under more restrictive conditions and a particular choice of generating set T.