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 $G_1 {\dot{\vee }} G_2$, $G_1 {\underline{\vee }} G_2$, $G_1 \langle \text {v} \rangle G_2$ and $G_1 \langle \text {e} \rangle G_2$ 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 $G_1 {\dot{\vee }} G_2$, $G_1 {\underline{\vee }} G_2$, $G_1 \langle \text {v} \rangle G_2$ and $G_1 \langle \text {e} \rangle G_2$ 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.