The maximal energy of classes of integral circulant graphs

Abstract

The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs, which can be characterized by their vertex count n and a set D of divisors of n in such a way that they have vertex set Zn and edge set {{a,b}:a,b∈Zn,gcd(a−b,n)∈D}. For a fixed prime power n=ps and a fixed divisor set size |D|=r, we analyse the maximal energy among all matching integral circulant graphs. Let pa1<pa2<⋯<par be the elements of D. It turns out that the differences di=ai+1−ai between the exponents of an energy maximal divisor set must satisfy certain balance conditions: (i) either all di equal q:=s−1r−1, or at most the two differences [q] and [q+1] may occur; (ii) there are rules governing the sequence d1,…,dr−1 of consecutive differences. For particular choices of s and r these conditions already guarantee maximal energy and its value can be computed explicitly.