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In this paper, we introduce two new graph operations, namely, the subdivisionvertex-edge neighbourhood vertex-corona and the subdivision vertex-edgeneighbourhood edge-corona on graphs $G_1$, $G_2$ and $G_3$, and the resultinggraphs are denoted by $G_1^S\bowtie (G_2^V\cup G_3^E)$ and$G_1^S\diamondsuit(G_2^V\cup G_3^E)$, respectively. Whereafter, the normalizedLaplacian spectra of $G_1^S\bowtie (G_2^V\cup G_3^E)$ and$G_1^S\diamondsuit(G_2^V\cup G_3^E)$ are respectively determined in terms ofthe corresponding normalized Laplacian spectra of the connected regular graphs$G_{1}$, $G_{2}$ and $G_{3}$, which extend the corresponding results of [A.Das, P. Panigrahi, Linear Multil. Algebra, 2017, 65(5): 962-972]. Asapplications, these results enable us to construct infinitely many pairs ofnormalized Laplacian cospectral graphs. Moreover, we also give the number ofthe spanning trees, the multiplicative degree-Kirchhoff index and Kemeny'sconstant of $G_1^S\bowtie (G_2^V\cup G_3^E)$ (resp.$G_1^S\diamondsuit(G_2^V\cup G_3^E)$).

The normalized Laplacian spectra of subdivision vertex-edge neighbourhood vertex(edge)-corona for graphs