THE EIGENVALUES AND ENERGY OF INTEGRAL CIRCULANT GRAPHS

A graph is called circulant if it is a Cayley graph on a cyclic group, i.e. its adjacency matrix is circulant. Let D be a set of positive, proper divisors of the integer n > 1. The integral circulant graph ICG(n) (D) has the vertex set Z(n) and the edge set E(ICG(n) (D)) = {{a, b}; gcd(a - b, n) is an element of D}. Let n = p(1)p(2) " " " pkm, where p(1), p(2), center dot center dot center dot, pk are distinct prime numbers and gcd(p(1)p(2) center dot center dot center dot pk, m) = 1. The open problem posed in paper [A. Ilie, The energy of unitary Cayley graphs, Linear Algebra Appl., 431 (2009) 1881-1889] about calculating the energy of an arbitrary integral circulant ICG(n)(D) is completely solved in this paper, where D = {p(1), p(2) center dot center dot center dot,pk}.