On the $ A_{\alpha^-} $-spectra of graphs and the relation between $ A_{\alpha} $- and $ A_{\alpha^-} $-spectra

Abstract

<abstract><p>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</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ A_{\alpha}(G) = \alpha D (G) + (1 - \alpha)A (G). $\end{document} </tex-math></disp-formula></p> <p>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</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ A_{\alpha^-}(G) = \alpha D (G) + (\alpha-1)A (G), \, \, \, 0 \leq \alpha \leq 1. $\end{document} </tex-math></disp-formula></p> <p>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 &lt; \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\}. $</p></abstract>