THE EIGENVALUES AND ENERGY OF INTEGRAL CIRCULANT GRAPHS

Abstract. 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 ICGn(D) has the vertex set Zn and the edge set E(ICGn(D)) = ffa; bg; gcd(a ?? b; n) 2 Dg. Let n = p1p2 ... pkm, where p1; p2; ... ; pk are distinct prime numbers and gcd(p1p2 ... pk;m) = 1. The open problem posed in paper [A. Ilic, The energy of unitary Cayley graphs, Linear Algebra Appl., 431 (2009) 1881{1889] about calculating the energy of an arbitrary integral circulant ICGn(D) is completely solved in this paper, where D = fp1; p2; : : : ; pkg.