The Laplacian energy of random graphs

TL;DR

This paper investigates the Laplacian energy of random graphs, establishing bounds and demonstrating that a previously conjectured inequality holds for almost all graphs.

Contribution

It provides bounds on Laplacian energy for random graphs and proves the conjecture holds for nearly all graphs, extending prior work on specific graph classes.

Findings

Bounds on Laplacian energy for random graphs

Counterexamples to the conjecture exist, but it holds for bipartite graphs

The conjecture is true for almost all graphs

Abstract

Gutman {\it et al.} introduced the concepts of energy $\En(G)$ and Laplacian energy $\EnL(G)$ for a simple graph $G$, and furthermore, they proposed a conjecture that for every graph $G$, $\En(G)$ is not more than $\EnL(G)$. Unfortunately, the conjecture turns out to be incorrect since Liu {\it et al.} and Stevanovi\'c {\it et al.} constructed counterexamples. However, So {\it et al.} verified the conjecture for bipartite graphs. In the present paper, we obtain, for a random graph, the lower and upper bounds of the Laplacian energy, and show that the conjecture is true for almost all graphs.