  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.

Combinatorics