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 analyze 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:=r−1s−1, or at most the two differences
[q] and [q+1] may occur; %(for a certain d depending on r and s) (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.Comment: Discrete Applied Mathematics (2012