The $A_{\alpha }$-spectra of graph operations based on generalized (edge) corona

Abstract

Let $G,H_i$ be simple graphs with $n=|V(G)|$, $m=|E(G)|$ and $i=1,2,\dots ,n(m)$. The generalized corona, denoted $G\tilde{o}{\wedge }_{i=1}^n{H}_i$, is the graph obtained by taking one copy of graphs $G,H_1,\dots ,H_n$ and joining the $i$th vertex of $G$ to every vertex of $H_i$ for $1\le i\le n$. The generalized edge corona, denoted by $G[H_i{]}_1^m$, is the graph obtained by taking one copy of graphs $G,H_1,\dots ,H_m$ and then joining two end-vertices of the $i$th edge of $G$ to every vertex of $H_i$ for $1\le i\le m$. For any real $\alpha \in [0,1]$, the matrix $A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the degree matrix of a graph $G$, respectively. In this paper, we obtain the $A_{\alpha }$-characteristic polynomial of $G\tilde{o}{\wedge }_{i=1}^n{H}_i$, which extends some known results. Meanwhile, we determine the $A_{\alpha }$-characteristic polynomial of $G[H_i{]}_1^m$ and get the $A_{\alpha }$-spectrum of $G[H_i{]}_1^m$ when $G$ and $H_i$ are regular graphs for $1\le i\le m$. As an application of the above conclusions, we construct infinitely many pairs of non-regular $A_{\alpha }$-cospectral graphs.