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 $\cal D$ of divisors of $n$ in such a way that they have vertex set $\mathbb{Z}_n$ and edge set ${{a,b}: a,b\in\mathbb{Z}_n, \gcd(a-b,n)\in {\cal D}}$. For a fixed prime power $n=p^s$ and a fixed divisor set size $|{\cal D}| =r$, we analyze the maximal energy among all matching integral circulant graphs. Let $p^{a_1} < p^{a_2} < ... < p^{a_r}$ be the elements of ${\cal D}$. It turns out that the differences $d_i=a_{i+1}-a_{i}$ between the exponents of an energy maximal divisor set must satisfy certain balance conditions: (i) either all $d_i$ equal $q:=\frac{s-1}{r-1}$, 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 $d_1,...,d_{r-1}$ of consecutive differences. For particular choices of $s$ and $r$ these conditions already guarantee maximal energy and its value can be computed explicitly.