The normalized Laplacian spectra of the double corona based on $R$-graph

For simple graphs $G$, $G_1$ and $G_2$, we denote their double corona based on $R$-graph by $G^{(R)}\otimes \{G_1,G_2\}$. This paper determines the normalized Laplacian spectrum of $G^{(R)}\otimes \{G_1,G_2\}$ in terms of these of $G$, $G_1$ and $G_2$ whenever $G$, $G_1$ and $G_2$ are regular. The obtained result reduces to the normalized Laplacian spectra of the $R$-vertex corona $G^{(R)}\odot G_1$ and $R$-edge corona $G^{(R)}⊝G_2$ by choosing $G_2$ or $G_1$ as a null-graph, respectively. Finally, applying the results of the paper, we construct infinitely many pairs of normalized Laplacian cospectral graphs.