The $T$-graph $T(G)$ of a graph $G$ is the graph whose vertices are the vertices and edges of $G$, with two vertices of $T(G)$ are adjacent if and only if the corresponding elements of $G$ are adjacent or incident. In this paper, we determine the adjacency and Laplacian spectra of $T$-vertex neighborhood corona and $T$-edge neighborhood corona of a connected regular graph with an arbitrary regular graph in terms of their eigenvalues. Moreover, applying these results we construct some non-regular $A$-cospectral and $L$-cospectral graphs.