On the $$A_{\alpha }$$-Spectra of Some Join Graphs

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