We consider two types of joins of graphs G1β and G2β, G1ββ»G2β - the Neighbors Splitting Join and G1β=β¨βG2β - 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 G1β and G2β are regular,
we compute the adjacency spectrum, the Laplacian spectrum, the signless
Laplacian spectrum of G1β=β¨βG2β and the normalized
Laplacian spectrum of G1ββ»G2β and G1β=β¨βG2β.
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