On the $A_{\alpha }$-spectra of some join graphs

Abstract

Let $G$ be a simple, connected graph and let $A(G)$ be the adjacency matrix of $G$. If $D(G)$ is the diagonal matrix of the vertex degrees of $G$, then for every real $\alpha \in [0,1]$, the matrix $A_{\alpha }(G)$ is defined 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 $G$. Let $G_1\dot{\vee }{G}_2$, $G_1\underline{\vee }{G}_2$, $G_1⟨\text{v}⟩G_2$ and $G_1⟨\text{e}⟩G_2$ denote the subdivision-vertex join, subdivision-edge join, $R$-vertex join and $R$-edge join of two graphs $G_1$ and $G_2$, respectively. In this paper, we compute the $A_{\alpha }$-spectra of $G_1\dot{\vee }{G}_2$, $G_1\underline{\vee }{G}_2$, $G_1⟨\text{v}⟩G_2$ and $G_1⟨\text{e}⟩G_2$ for a regular graph $G_1$ and an arbitrary graph $G_2$ in terms of their $A_{\alpha }$-eigenvalues. As an application of these results, we construct infinitely many pairs of $A_{\alpha }$-cospectral graphs.