Integral circulant graphs of prime power order with maximal energy

TL;DR

This paper investigates the maximum possible energy of integral circulant graphs of prime power order, providing bounds, constructions, and characterizations of hyperenergetic cases using convex optimization techniques.

Contribution

It introduces bounds and constructions for maximal energy integral circulant graphs of prime power order, and characterizes hyperenergetic instances with specific divisor set properties.

Findings

Maximal energy lies between s(p - 1)p^(s-1) and twice this value.

Constructed divisor sets achieve energies within these bounds.

Characterized hyperenergetic graphs and their divisor set topologies.

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 in Zn; gcd(a - b, n) in D}. Using tools from convex optimization, we study the maximal energy among all integral circulant graphs of prime power order ps and varying divisor sets D. Our main result states that this maximal energy approximately lies between s(p - 1)p^(s-1) and twice this value. We construct suitable divisor sets for which the energy lies in this interval. We also characterize hyperenergetic integral circulant graphs of prime power order and exhibit an interesting topological property of their divisor sets.