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

Let G be a graph on n vertices and m edges. For \alpha \in [۰,۱], the A_{\alpha}-matrix of G is defined as A_{\alpha}(G) = \alpha D(G) + (۱- \alpha) A(G), where A(G) is the adjacency matrix and D(G) is the degree diagonal matrix of G. If \rho_۱ \geq \rho_۲ \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=۱}^{n} |\rho_i -\frac{۲\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.

کلیدواژه ها:

Graphs ، A_{\alpha}-matrix ، A_{\alpha}-eigenvalues ، A_{\alpha}-energy ، bounds