New constructions of non-regular cospectral graphs

We consider two types of joins of graphs $G_1$ and $G_2$, $G_1⊻G_2$ - the Neighbors Splitting Join and $G_1{G}_2$ - the Non Neighbors Splitting Join, and compute the adjacency characteristic polynomial, the Laplacian characteristic polynomial and the signless Laplacian characteristic polynomial of these joins. When $G_1$ and $G_2$ are regular, we compute the adjacency spectrum, the Laplacian spectrum, the signless Laplacian spectrum of $G_1{G}_2$ and the normalized Laplacian spectrum of $G_1⊻G_2$ and $G_1{G}_2$. We use these results to construct non regular, non isomorphic graphs that are cospectral with respect to the four matrices: adjacency, Laplacian , signless Laplacian and normalized Laplacian.