Normalized Laplacian spectrum of some subdivision-joins and R-joins of two regular graphs

In this paper we determine the full normalized Laplacian spectrum of the subdivision-vertex join, subdivision-edge join, R-vertex join, and R-edge join of two regular graphs in terms of the normalized Laplacian eigenvalues of the graphs. Moreover, applying these results we find non-regular normalized Laplacian cospectral graphs. Keywords: Normalized Laplacian matrix, Subdivision-vertex join, Subdivision-edge join, R-vertex join, R-edge joi