On the Aα-Spectra of Some Join Graphs

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 is an element of [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)boolean OR G(2), G(1)boolean OR G(2), G(1)< v > G(2) and G(1)< 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)boolean OR G(2), G(1)boolean OR G(2), G(1)< v > G(2) and G(1)< 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.