# NOTE ON DELETING A VERTEX AND WEAK INTERLACING OF THE LAPLACIAN SPECTRUM

@article{Lotker2007NOTEOD, title={NOTE ON DELETING A VERTEX AND WEAK INTERLACING OF THE LAPLACIAN SPECTRUM}, author={Zvi Lotker}, journal={Electronic Journal of Linear Algebra}, year={2007}, volume={16}, pages={6} }

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 λi − 1 ≤ λ v ≤ λi+1, where λi is the ith smallest eigenvalues of the Laplacian of the original graph and λ 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 λn ,i f λ2 >α n.

## 14 Citations

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