Exploring new upper and lower bounds for the $A_{\alpha}$-energy of graphs

TL;DR

This paper introduces new bounds for the $A_{\alpha}$-energy of graphs, providing sharper estimates and relations to other graph energies, enhancing understanding of spectral graph properties.

Contribution

It presents novel sharp upper and lower bounds for $A_{\alpha}$-energy, compares them with existing results, and explores their connections to other graph energies.

Findings

New sharp bounds for $A_{\alpha}$-energy established.

Improved estimates over previous bounds demonstrated.

Relations between $A_{\alpha}$-energy and other graph energies derived.

Abstract

Let $G$ be a graph on $n$ vertices and $m$ edges. For $\alpha \in [0,1]$, the $A_{\alpha}$-matrix of $G$ is defined as $A_{\alpha}(G) = \alpha D(G) + (1- \alpha) A(G)$, where $A(G)$ is the adjacency matrix and $D(G)$ is the degree diagonal matrix of $G$. If $\rho_1 \geq \rho_2 \ldots \geq \rho_n$ are the eigenvalues of $A_{\alpha}(G)$, the $A_{\alpha}$-energy of $G$ is defined as $E_{A_{\alpha}}(G) = \sum_{i=1}^{n} |\rho_i -\frac{2\alpha m}{n}|$. In this paper, we present novel upper and lower bounds for $E_{A_\alpha}(G)$ in terms of standard graph invariants, showing that each bound is sharp and identifying the specific graphs attaining them. For selected bounds, we provide brief comparative analysis with existing results, observing improved estimates. Furthermore, we establish new relations between $E_{A_\alpha}(G)$ and other well known graph energies, including adjacency, Laplacian, as well as the adjacency energy of the line graph.