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 \(G^{(R)}\circleddash{G_2}\) 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.