Note on deleting a vertex and weak interlacing of the Laplacian spectrum

The question of what happens to the eigenvalues of the Laplacian of a graph when we delete a vertex is addressed. It is shown that λ_i - 1 ≤_i^v λ_i+1, where λ_i is the ith smallest eigenvalues of the Laplacian of the original graph and λ_i^v is the ith smallest eigenvalues of the Laplacian of the graph G[V - v]; i.e., the graph obtained after removing the vertex v. It is shown that the average number of leaves in a random spanning tree ℱ(G) > 2|E|e^-1/α/λ_n, if λ₂ > αn.