Consider the Laplacian and signless Laplacian spectrum of a graph G of order n, with k pairwise co-neighbor vertices. We prove that the number of shared neighbors is a Laplacian and a signless Laplacian eigenvalue of G with multiplicity at least k − 1. Additionally, considering a connected graph GkGk with a vertex set defined by the k pairwise co-neighbor vertices of G, the Laplacian spectrum of GkGk, obtained from G adding the edges of GkGk, includes l+βl+β for each nonzero Laplacian eigenvalue ββ of GkGk. The Laplacian spectrum of G overlaps the Laplacian spectrum of GkGk in at least n − k + 1 places.