Regular graphs with maximal energy per vertex

TL;DR

This paper investigates the maximum energy per vertex in regular graphs, establishing tight bounds and characterizing extremal graphs, including constructions that approach these bounds asymptotically.

Contribution

It provides the first tight upper bounds for energy per vertex in regular graphs and characterizes extremal cases using incidence graphs of projective planes.

Findings

Upper bounds for energy per vertex in k-regular graphs.

Characterization of extremal graphs as unions of incidence graphs.

Construction of subgraphs approaching the upper bound asymptotically.

Abstract

We study the energy per vertex in regular graphs. For every k, we give an upper bound for the energy per vertex of a k-regular graph, and show that a graph attains the upper bound if and only if it is the disjoint union of incidence graphs of projective planes of order k-1 or, in case k=2, the disjoint union of triangles and hexagons. For every k, we also construct k-regular subgraphs of incidence graphs of projective planes for which the energy per vertex is close to the upper bound. In this way, we show that this upper bound is asymptotically tight.