Let $ G $ be a graph with adjacency matrix $ A(G) $, and let $ D(G) $ be the diagonal matrix of the degrees of $ G $. For any real number $ \alpha\in [0, 1] $, Nikiforov defined the $ A_{\alpha} $-matrix of $ G $ as
$ A_{\alpha}(G) = \alpha D (G) + (1 - \alpha)A (G). $
The eigenvalues of the matrix $ A_{\alpha}(G) $ form the $ A_{\alpha} $-spectrum of $ G $. The $ A_{\alpha} $-spectral radius of $ G $ is the largest eigenvalue of $ A_{\alpha}(G) $ denoted by $ \rho_\alpha(G) $. In this paper, we propose the $ A_{\alpha^-} $-matrix of $ G $ as
$ A_{\alpha^-}(G) = \alpha D (G) + (\alpha-1)A (G), \, \, \, 0 \leq \alpha \leq 1. $
Let the $ A_{\alpha^-} $-spectral radius of $ G $ be denoted by $ \lambda_{\alpha^-}(G) $, and let $ S^{\alpha}_{\beta}(G) $ and $ S^{\alpha^-}_{\beta}(G) $ be the sum of the $ \beta^{th} $ powers of the $ A_{\alpha} $ and $ A_{\alpha^-} $ eigenvalues of $ G $, respectively. We determine the $ A_{\alpha^-} $-spectra of some graphs and obtain some bounds of the $ A_{\alpha^-} $-spectral radius. Moreover, we establish a relationship between the $ A_{\alpha} $-spectral radius and $ A_{\alpha^-} $-spectral radius. Indeed, for $ \alpha\in(\frac{1}{2}, 1) $, we show that $ \lambda_{\alpha^-}\leq \rho_\alpha $, and we prove that if $ G $ is connected, then the equality holds if and only if $ G $ is bipartite. Employing this relation, we obtain some upper bounds of $ \lambda_{\alpha^-}(G) $, and we prove that the $ A_{\alpha^-} $-spectrum and $ A_\alpha $-spectrum are equal if and only if $ G $ is a bipartite connected graph. Furthermore, we generalize the relation established by S. Akbari et al. in $ (2010) $ as follows: for $ \alpha\in[\frac{1}{2}, 1) $, if $ \, \, \, 0 < \beta\leq 1 $ or $ \, 2\leq\beta\leq 3 $, then $ S^{\alpha}_{\beta}(G)\geq S^{\alpha^-}_{\beta}(G), $ and if $ \, 1\leq\beta\leq 2 $, then $ S^{\alpha}_{\beta}(G)\leq S^{\alpha^-}_{\beta}(G), $ where the equality holds if and only if $ G $ is a bipartite graph such that $ \beta \notin \{1, 2, 3\}. $
Citation: Wafaa Fakieh, Zakeiah Alkhamisi, Hanaa Alashwali. On the $ A_{\alpha^-} $-spectra of graphs and the relation between $ A_{\alpha} $- and $ A_{\alpha^-} $-spectra[J]. AIMS Mathematics, 2024, 9(2): 4587-4603. doi: 10.3934/math.2024221
Let $ G $ be a graph with adjacency matrix $ A(G) $, and let $ D(G) $ be the diagonal matrix of the degrees of $ G $. For any real number $ \alpha\in [0, 1] $, Nikiforov defined the $ A_{\alpha} $-matrix of $ G $ as
$ A_{\alpha}(G) = \alpha D (G) + (1 - \alpha)A (G). $
The eigenvalues of the matrix $ A_{\alpha}(G) $ form the $ A_{\alpha} $-spectrum of $ G $. The $ A_{\alpha} $-spectral radius of $ G $ is the largest eigenvalue of $ A_{\alpha}(G) $ denoted by $ \rho_\alpha(G) $. In this paper, we propose the $ A_{\alpha^-} $-matrix of $ G $ as
$ A_{\alpha^-}(G) = \alpha D (G) + (\alpha-1)A (G), \, \, \, 0 \leq \alpha \leq 1. $
Let the $ A_{\alpha^-} $-spectral radius of $ G $ be denoted by $ \lambda_{\alpha^-}(G) $, and let $ S^{\alpha}_{\beta}(G) $ and $ S^{\alpha^-}_{\beta}(G) $ be the sum of the $ \beta^{th} $ powers of the $ A_{\alpha} $ and $ A_{\alpha^-} $ eigenvalues of $ G $, respectively. We determine the $ A_{\alpha^-} $-spectra of some graphs and obtain some bounds of the $ A_{\alpha^-} $-spectral radius. Moreover, we establish a relationship between the $ A_{\alpha} $-spectral radius and $ A_{\alpha^-} $-spectral radius. Indeed, for $ \alpha\in(\frac{1}{2}, 1) $, we show that $ \lambda_{\alpha^-}\leq \rho_\alpha $, and we prove that if $ G $ is connected, then the equality holds if and only if $ G $ is bipartite. Employing this relation, we obtain some upper bounds of $ \lambda_{\alpha^-}(G) $, and we prove that the $ A_{\alpha^-} $-spectrum and $ A_\alpha $-spectrum are equal if and only if $ G $ is a bipartite connected graph. Furthermore, we generalize the relation established by S. Akbari et al. in $ (2010) $ as follows: for $ \alpha\in[\frac{1}{2}, 1) $, if $ \, \, \, 0 < \beta\leq 1 $ or $ \, 2\leq\beta\leq 3 $, then $ S^{\alpha}_{\beta}(G)\geq S^{\alpha^-}_{\beta}(G), $ and if $ \, 1\leq\beta\leq 2 $, then $ S^{\alpha}_{\beta}(G)\leq S^{\alpha^-}_{\beta}(G), $ where the equality holds if and only if $ G $ is a bipartite graph such that $ \beta \notin \{1, 2, 3\}. $
| [1] |
V. Nikiforov, Merging the $A$- and $Q$-spectral theories, Appl. Anal. Discrete Math., 11 (2017), 81–107. https://doi.org/10.2298/AADM1701081N doi: 10.2298/AADM1701081N
|
| [2] |
V. Nikiforov, G. Pastén, O. Rojo, R. L. Soto, On the $A_\alpha$-spectra of trees, Linear Algebra Appl., 520 (2017), 286–305. https://doi.org/10.1016/j.laa.2017.01.029 doi: 10.1016/j.laa.2017.01.029
|
| [3] |
H. Lin, J. Xue, J. Shu, On the $A_{\alpha}$-spectra of graphs, Linear Algebra Appl., 556 (2018), 210–219. https://doi.org/10.1016/j.laa.2018.07.003 doi: 10.1016/j.laa.2018.07.003
|
| [4] |
H. Guo, B. Zhou, On the $\alpha$-spectral radius of graphs, Appl. Anal. Discrete Math., 14 (2020), 431–458. https://doi.org/10.2298/AADM180210022G doi: 10.2298/AADM180210022G
|
| [5] |
S. Guo, R. Zhang, Ordering graphs by their largest (least) $A_\alpha$-eigenvalues, Linear Multilinear Algebra, 70 (2022), 7049–7056. https://doi.org/10.1080/03081087.2021.1981811 doi: 10.1080/03081087.2021.1981811
|
| [6] |
M. Basunia, I. Mahato, M. R. Kannan, On the $A_\alpha$-spectra of some join graphs, Bull. Malays. Math. Sci. Soc., 44 (2021), 4269–4297. https://doi.org/10.1007/s40840-021-01166-z doi: 10.1007/s40840-021-01166-z
|
| [7] |
H. Lin, X. Huang, J. Xue, A note on the $A_{\alpha}$-spectral radius of graphs, Linear Algebra Appl., 557 (2018), 430–437. https://doi.org/10.1016/j.laa.2018.08.008 doi: 10.1016/j.laa.2018.08.008
|
| [8] |
J. Wang, J. Wang, X. Liu, F. Belardo, Graphs whose $A_{\alpha}$-spectral radius does not exceed 2, Discuss. Math. Graph Theory, 40 (2020), 677–690. https://doi.org/10.7151/dmgt.2288 doi: 10.7151/dmgt.2288
|
| [9] | A. E. Brouwer, W. H. Haemers, Spectra of graphs, Springer Science & Business Media, 2012. https://doi.org/10.1007/978-1-4614-1939-6 |
| [10] |
S. Akbari, E. Ghorbani, J. H. Koolen, M. R. Oboudi, On sum of powers of the Laplacian and signless Laplacian eigenvalues of graphs, Electron. J. Comb., 2017 (2010), R115. https://doi.org/10.37236/387 doi: 10.37236/387
|
| [11] |
R. A. Horn, C. R. Johnson, Matrix analysis, 2 Eds., Cambridge University Press, 2012. https://doi.org/10.1017/CBO9781139020411 doi: 10.1017/CBO9781139020411
|
| [12] |
W. So, Commutativity and spectra of Hermitian matrices, Linear Algebra Appl., 212-213 (1994), 121–129. https://doi.org/10.1016/0024-3795(94)90399-9 doi: 10.1016/0024-3795(94)90399-9
|
| [13] |
J. Shu, Y. Hong, K. Ren, A sharp upper bound on the largest eigenvalue of the Laplacian matrix of a graph, Linear Algebra Appl., 347 (2002), 123–129. https://doi.org/10.1016/S0024-3795(01)00548-1 doi: 10.1016/S0024-3795(01)00548-1
|
| [14] |
L. You, M. Yan, W. So, W. Xi, On the spectrum of an equitable quotient matrix and its application, Linear Algebra Appl., 577 (2019), 21–40. https://doi.org/10.1016/j.laa.2019.04.013 doi: 10.1016/j.laa.2019.04.013
|
| [15] |
R. Merris, Laplacian matrices of graphs: a survey, Linear Algebra Appl., 197-198 (1994), 143–176. https://doi.org/10.1016/0024-3795(94)90486-3 doi: 10.1016/0024-3795(94)90486-3
|
| [16] |
S. Pirzada, B. A. Rather, R. Ul Shaban, T. A. Chishti, On the sum of the powers of $A_\alpha$ eigenvalues of graphs and $A_\alpha$-energy like invariant, Bol. Soc. Paran. Mat., 40 (2022), 1–12. https://doi.org/10.5269/bspm.52469 doi: 10.5269/bspm.52469
|
| [17] | R. G. Bartle, The elements of real analysis, 2 Eds., John Wiley & Sons, Inc., 1976. |
| [18] | N. Biggs, Algebraic graph theory, 2 Eds., Cambridge University Press, 1993. https://doi.org/10.1017/CBO9780511608704 |
Wafaa Fakieh, Zakeiah Alkhamisi, Hanaa Alashwali. On the $ A_{\alpha^-} $-spectra of graphs and the relation between $ A_{\alpha} $- and $ A_{\alpha^-} $-spectra[J]. AIMS Mathematics, 2024, 9(2): 4587-4603. doi: 10.3934/math.2024221
DownLoad: