A lower bound of the energy of non-singular graphs in terms of average degree

TL;DR

This paper investigates a lower bound on the energy of non-singular graphs, proposing a stronger conjecture relating energy to average degree, and verifies it for specific classes of graphs.

Contribution

It introduces a new conjecture linking graph energy to average degree and proves it for bipartite, planar, and certain sparse graphs.

Findings

Conjecture holds for bipartite graphs.

Conjecture holds for planar graphs.

Conjecture holds for graphs with low average degree.

Abstract

Let $G$ be a graph of order $n$ with adjacency matrix $A(G)$. The \textit{energy} of graph $G$, denoted by $\mathcal{E}(G)$, is defined as the sum of absolute value of eigenvalues of $A(G)$. It was conjectured that if $A(G)$ is non-singular, then $\mathcal{E}(G)\geq\Delta(G)+\delta(G)$. In this paper we propose a stronger conjecture as for $n \geq 5$, $\mathcal{E}(G)\geq n-1+ d$, where $d$ is the average degree of $G$. Here, we show that conjecture holds for bipartite graphs, planar graphs and for the graphs with $d \leq n-2\ln n -3$