Extremal energies of integral circulant graphs via multiplicativity

Abstract The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. Integral circulant graphs can be characterised by their order n and a set D of positive divisors of n in such a way that they have vertex set Z / n Z and edge set { ( a , b ) : a , b ∈ Z / n Z , gcd ( a - b , n ) ∈ D } . Among integral circulant graphs of fixed prime power order p s , those having minimal energy E min p s or maximal energy E max p s , respectively, are known. We study the energy of integral circulant graphs of arbitrary order n with so-called multiplicative divisor sets. This leads to good bounds for E min n and E max n as well as conjectures concerning the true value of E min n .