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}
}
  • Zvi Lotker
  • Published 2007
  • Mathematics
  • Electronic Journal of Linear Algebra
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. 

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The Cauchy interlace theorem states that the eigenvalues of a Hermitian matrix A of order n are interlaced with those of any principal submatrix of ordern − 1.

Algebraic Graph Theory

The Laplacian of a Graph and Cuts and Flows are compared to the Rank Polynomial.

Matrix Computations. 2nd ed

  • Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume
  • 1989