The exact maximal energy of integral circulant graphs with prime power order

TL;DR

This paper precisely determines the maximum energy of integral circulant graphs with prime power order by characterizing divisor sets that maximize their spectral sum, advancing understanding of graph energy in algebraic graph theory.

Contribution

It provides a complete characterization of divisor sets that maximize the energy of integral circulant graphs with prime power order, enabling exact computation of their maximum energy.

Findings

Derived explicit formulas for maximal energy of graphs with prime power order.

Identified divisor sets that achieve maximum energy.

Computed the maximum energy for all prime power orders.

Abstract

The energy of a graph was introduced by {\sc Gutman} in 1978 as the sum of the absolute values 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 $\mathbb{Z}/n\mathbb{Z}$ and edge set $\{\{a,b\}:\, a,b\in\mathbb{Z}/n\mathbb{Z},\, \gcd(a-b,n)\in {\cal D}\}$. Given an arbitrary prime power $p^s$, we determine all divisor sets maximising the energy of an integral circulant graph of order $p^s$. This enables us to compute the maximal energy $\Emax{p^s}$ among all integral circulant graphs of order $p^s$.