On the Automorphism Group of Integral Circulant Graphs

The integral circulant graph Xn(D) has the vertex set Zn = {0, 1, 2, . . . , n − 1} and vertices a and b are adjacent, if and only if gcd(a − b, n) ∈ D, where D = {d1, d2, . . . , dk} is a set of divisors of n. These graphs play an important role in modeling quantum spin networks supporting the perfect state transfer and also have applications in chemical graph theory. In this paper, we deal with the automorphism group of integral circulant graphs and investigate a problem proposed in [W. Klotz, T. Sander, Some properties of unitary Cayley graphs, Electr. J. Comb. 14 (2007), #R45]. We determine the size and the structure of the automorphism group of the unitary Cayley graph Xn(1) and the disconnected graph Xn(d). In addition, based on the generalized formula for the number of common neighbors and the wreath product, we completely characterize the automorphism groups Aut(Xn(1, p)) for n being a square-free number and p a prime dividing n, and Aut(Xn(1, p k)) for n being a prime power.