Open AccessOn the sum of the powers of $ A_\alpha $ eigenvalues of graphs and $ A_\alpha $-energy like invariant
Author(s) -
S. Pirzada,
Bilal A. Rather,
Rezwan Ul Shaban,
T. A. Chishti
Publication year - 2022
Publication title -
boletim da sociedade paranaense de matemática
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.347
H-Index - 15
eISSN - 2175-1188
pISSN - 0037-8712
DOI - 10.5269/bspm.52469
Subject(s) - combinatorics , eigenvalues and eigenvectors , graph , invariant (physics) , mathematics , simple graph , beta (programming language) , alpha (finance) , discrete mathematics , physics , mathematical physics , quantum mechanics , computer science , construct validity , statistics , programming language , psychometrics
For a connected simple graph $ G $ with $ A_{\alpha} $ eigenvalues $ \rho_{1}\geq\rho_{2}\geq\dots\geq\rho_{n} $ and a real number $\beta $, let $ S_{\beta}^{\alpha}(G) =\sum\limits_{i=1}^{n}\rho_{i}^{\beta}$ be the sum of the $ \beta^{th} $ powers of the $ A_{\alpha} $ eigenvalues of graph $ G $. In this paper, we obtain various bounds for the graph invariant $ S_{\beta}^{\alpha}(G) $ in terms of different graph parameters. As a consequence, we obtain the bounds for the quantity $ IE^{A_{\alpha}}(G)= S_{\frac{1}{2}}^{\alpha}(G),$ the $ A_{\alpha} $ energy-like invariant of the graph $ G .$