New constructions of non-regular cospectral graphs

Abstract: We consider two types of joins of graphs $G_{1}$ and $G_{2}$, $G_{1}\veebar G_{2}$ - the Neighbors Splitting Join and $G_{1}\underset{=}{\lor}G_{2}$ - 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 $G_{1}\underset{=}{\lor}G_{2}$ and the normalized Laplacian spectrum of $G_{1}\veebar G_{2}$ and $G_{1}\underset{=}{\lor}G_{2}$. 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.