The $B_{\alpha}$-spectra of graph product

Abstract

For a graph $G$, let $A(G)$ and $L(G)$ denote the adjacency matrix and the Laplacian matrix of $G$, respectively. Define
$$B_{\alpha}(G)=\alpha A(G)+(1-\alpha)L(G), \mbox{ for any real $\alpha \in[0,1]$}.$$
The collection of eigenvalues of $B_{\alpha}(G)$ together with multiplicities is called the $B_\alpha$-spectrum of $G$. Let $G\Box H$, $G[H]$, $G\times H$ and $G\oplus H$ be the Cartesian product, lexicographic product, directed product and strong product of graphs $G$ and $H$, respectively. In this paper, we determine the $B_{\alpha}$-spectrum of $G\Box H$ for arbitrary graphs $G$ and $H$, and $G[H]$ for arbitrary graph $G$ and regular graph $H$. Furthermore, we consider the $B_{\alpha}$-spectrum of the generalized lexicographic product $G\left[H_1, H_2, \ldots, H_n\right]$ for $n$-vertex graph $G$ and regular graphs $H_i$'s. Finally, for any real $\alpha \in[\frac{1}{2},1]$, we obtain the spectral radii of $B_{\alpha}(G\times H)$ and $B_{\alpha}(G \oplus H)$ for arbitrary graph $G$ and regular graph $H$.