  In this paper, we introduce two new graph operations, namely, the subdivision
vertex-edge neighbourhood vertex-corona and the subdivision vertex-edge
neighbourhood edge-corona on graphs $G_1$, $G_2$ and $G_3$, and the resulting
graphs 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 normalized
Laplacian 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 of
the 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]. As
applications, these results enable us to construct infinitely many pairs of
normalized Laplacian cospectral graphs. Moreover, we also give the number of
the spanning trees, the multiplicative degree-Kirchhoff index and Kemeny's
constant of $G_1^S\bowtie (G_2^V\cup G_3^E)$ (resp.
$G_1^S\diamondsuit(G_2^V\cup G_3^E)$).
