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

Articles in Press

Document Type : Research Paper

Authors

Department of Mathematics, Indian Institute of Technology Kharagpur, India

10.48308/cgasa.2026.242533.1585

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.

Keywords

Main Subjects

References

[1] Abreu, N., Cardoso, D.M., Gutman, I., Martins, E.A., and Robbiano, M., Bounds for the signless Laplacian energy, Linear Algebra Appl. 435(10) (2011), 2365-2374.
[2] Alhevaz, A., Baghipur, M., and Ganie, H.A., Bounds for the spectral radius of the Aα-matrix of graphs, Indian J. Pure Appl. Math. 55(1) (2024), 298-309.
[3] Basunia, M., Mahato, I., and Kannan, M.R., On the Aα-Spectra of Some Join Graphs, Bull. Malays. Math. Sci. Soc. 44(6) (2021), 4269-4297.
[4] Cvetkovi´c, D., Rowlinson, P., and Simi´c, S., “An Introduction to the Theory of Graph Spectra”, Cambridge University Press, 2010.
[5] Das, J. and Mahato, I., On the maximum Aα-spectral radius of unicyclic and bicyclic graphs with fixed girth or fixed number of pendant vertices, Comput. Appl. Math. 43(6) (2024), Article no. 347.
[6] Das, K.C. and Mojallal, S.A., On Laplacian energy of graphs, Discrete Math. 325 (2014), 52-64.
[7] Das, K.C. and Mojallal, S.A., Relation between signless Laplacian energy, energy of graph and its line graph, Linear Algebra Appl. 493 (2016), 91-107.
[8] de Caen, D., An upper bound on the sum of squares of degrees in a graph, Discrete Math. 185(1-3) (1998), 245-248.
[9] Fan, K., On a theorem of Weyl concerning eigenvalues of linear transformations. I, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 652-655.
[10] Fan, K., Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 760-766.
[11] Guo, H. and Zhou, B., On the α-spectral radius of graphs, Appl. Anal. Discrete Math. 14(2) (2020), 431-458.
[12] Gutman, I., The energy of a graph, Ber. Math.-Statist. Sekt. Forsch. Graz 103 (1978), 1-22.
[13] Gutman, I., Robbiano, M., Martins, E.A., Cardoso, D.M., Medina, L., and Rojo, O., Energy of line graphs, Linear Algebra Appl. 433(7) (2010), 1312-1323.
[14] Gutman, I. and Zhou, B., Laplacian energy of a graph, Linear Algebra Appl. 414(1) (2006), 29-37.
[15] Horn, R.A. and Johnson, C.R., gMatrix Analysish, Cambridge University Press, 2012.
[16] Lin, H., Xue, J., and Shu, J., On the Aα-spectra of graphs, Linear Algebra Appl. 556 (2018), 210-219.
[17] Lin, Z., On the sum of the largest Aα-eigenvalues of graphs, AIMS Math. 7(8) (2022), 15064-15074.
[18] Lin, Z., Miao, L., and Guo, S.G., Bounds on the Aα-spread of a graph, Electron. J. Linear Algebra 36 (2020), 214-227.
[19] MitrinoviLc, D.S., Analytic Inequalitiesh, Springer-Verlag, 1970.
[20] Nikiforov, V., Merging the A- and Q-spectral theories, Appl. Anal. Discrete Math. 11(1) (2017), 81-107.
[21] Nikiforov, V. and Rojo, O., A note on the positive semidefiniteness of Aα(G), Linear Algebra Appl. 519 (2017), 156-163.
[22] Pirzada, S., Rather, B.A., Ganie, H.A., and Shaban, R.U., On α-adjacency energy of graphs and Zagreb index, AKCE Int. J. Graphs Comb. 18(1) (2021), 39-46.
[23] Shaban, R.U., Imran, M., and Ganie, H.A., Bounds for the α-adjacency energy of a graph, J. Math. Inequal. 18(1) (2024), 127-141.
[24] So, W., Commutativity and spectra of Hermitian matrices, Linear Algebra Appl. 212/213 (1994), 121-129.
[25] Zhou, L., Li, D., Chen, Y., and Meng, J., Some bounds on the Aƒ¿-energy of graphs, Filomat 38(4) (2024), 1329-1341.
Categories and General Algebraic Structures with Applications

Articles in Press, Accepted Manuscript
Available Online from 06 June 2026

Files

Share

How to cite

Statistics