New results on the energy of integral circulant graphs

TL;DR

This paper investigates the energy of integral circulant graphs, providing modular characterizations, closed-form formulas, and identifying extremal graphs with minimal energy for even-sized graphs.

Contribution

It offers new modular results, general formulas for energy, and characterizes extremal minimal-energy graphs among integral circulant graphs.

Findings

Characterized energy modulo 4 for integral circulant graphs.

Derived general closed-form expressions for graph energy.

Identified extremal graphs with minimal energy when n is even.

Abstract

Circulant graphs are an important class of interconnection networks in parallel and distributed computing. Integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer as well. The integral circulant graph $\ICG_n (D)$ has the vertex set $Z_n = \{0, 1, 2, ..., n - 1\}$ and vertices $a$ and $b$ are adjacent if $\gcd(a-b,n)\in D$, where $D \subseteq \{d : d \mid n,\ 1\leq d<n\}$. These graphs are highly symmetric, have integral spectra and some remarkable properties connecting chemical graph theory and number theory. The energy of a graph was first defined by Gutman, as the sum of the absolute values of the eigenvalues of the adjacency matrix. Recently, there was a vast research for the pairs and families of non-cospectral graphs having equal energies. Following [R. B. Bapat, S. Pati, \textit{Energy of a graph is never an odd integer}, Bull. Kerala Math. Assoc. 1 (2004) 129--132.], we characterize the energy of integral circulant graph modulo 4. Furthermore, we establish some general closed form expressions for the energy of integral circulant graphs and generalize some results from [A. Ili\' c, \textit{The energy of unitary Cayley graphs}, Linear Algebra Appl. 431 (2009), 1881--1889.]. We close the paper by proposing some open problems and characterizing extremal graphs with minimal energy among integral circulant graphs with $n$ vertices, provided $n$ is even.