Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 25 1 2026 07 01 Exploring new upper and lower bounds for the $A_{\alpha}$-energy of graphs 107062 10.48308/cgasa.2026.242533.1585 EN Mainak Basunia Department of Mathematics, Indian Institute of Technology Kharagpur, India Pratima Panigrahi Department of Mathematics, Indian Institute of Technology Kharagpur, India Journal Article 2025 11 17 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. https://cgasa.sbu.ac.ir/article_107062_9f47c6429548a3dfe3fa5321692900b9.pdf