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 $i=1, 2, \ldots, n(m)$. The generalized corona, denoted $G\tilde{o}\wedge^{n}_{i=1} H_{i}$, is the graph obtained by taking one copy of graphs $G, H_{1},\ldots, H_{n}$ and joining the $i$th vertex of $G$ to every vertex of $H_{i}$ for $1 \leq i \leq n$. The generalized edge corona, denoted by $G[H_i]_1^m$, is the graph obtained by taking one copy of graphs $G, H_{1},\ldots, H_{m}$ and then joining two end-vertices of the $i$th edge of $G$ to every vertex of $H_{i}$ for $1 \leq i \leq m$. 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 $G\tilde{o}\wedge^{n}_{i=1} H_{i}$, 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.