On the normalized Laplacian spectra of some subdivision joins of two graphs

For two simple graphs $G_1$ and $G_2$, we denote the subdivision-vertex join and subdivision-edge join of $G_1$ and $G_2$ by $G_1\dot{\vee}G_2$ and $G_1\veebar G_2$, respectively. This paper determines the normalized Laplacian spectra of $G_1\dot{\vee}G_2$ and $G_1\veebar G_2$ in terms of these of $G_1$ and $G_2$ whenever $G_1$ and $G_2$ are regular. As applications, we construct some non-regular normalized Laplacian cospectral graphs. Besides we also compute the number of spanning trees and the degree-Kirchhoff index of $G_1\dot{\vee}G_2$ and $G_1\veebar G_2$ for regular graphs $G_1$ and $G_2$.