Remarks on the energy of regular graphs

TL;DR

This paper investigates the energy of regular graphs, demonstrating how near-regular graphs can be adjusted to regularity with minimal energy change, and constructs large regular graphs with high energy, providing new insights into their spectral properties.

Contribution

It introduces methods to modify near-regular graphs to regular ones with negligible energy change and constructs large regular graphs with nearly maximal energy.

Findings

Near-regular graphs can be made regular with negligible energy change

Existence of regular graphs with energy exceeding half of n^{3/2} for large n

Characterization of energy for almost all regular graphs

Abstract

The energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. This note is about the energy of regular graphs. It is shown that graphs that are close to regular can be made regular with a negligible change of the energy. Also a $k$-regular graph can be extended to a $k$-regular graph of a slightly larger order with almost the same energy. As an application, it is shown that for every sufficiently large $n,$ there exists a regular graph $G$ of order $n$ whose energy $\left\Vert G\right\Vert_{\ast}$ satisfies \[ \left\Vert G\right\Vert_{\ast}>\frac{1}{2}n^{3/2}-n^{13/10}. \] Several infinite families of graphs with maximal or submaximal energy are given, and the energy of almost all regular graphs is determined.