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

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

$$A_{{\alpha }^{-}}(G)=\alpha D(G)+(\alpha -1)A(G),0\le \alpha \le 1.$$

Let the $A_{{\alpha }^{-}}$-spectral radius of $G$ be denoted by ${\lambda }_{{\alpha }^{-}}(G)$, and let $S_{\beta }^{\alpha }(G)$ and $S_{\beta }^{{\alpha }^{-}}(G)$ 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 ${\lambda }_{{\alpha }^{-}}\le {\rho }_{\alpha }$, 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 $0<\beta \le 1$ or $2\le \beta \le 3$, then $S_{\beta }^{\alpha }(G)\ge S_{\beta }^{{\alpha }^{-}}(G),$ and if $1\le \beta \le 2$, then $S_{\beta }^{\alpha }(G)\le S_{\beta }^{{\alpha }^{-}}(G),$ where the equality holds if and only if $G$ is a bipartite graph such that $\beta \notin \{1,2,3\}.$