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

Zvi Lotker

doi:10.13001/1081-3810.1183

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

Zvi Lotker

Research output: Contribution to journal › Article › peer-review

14 Scopus citations

Abstract

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.

Original language	English
Pages (from-to)	68-72
Number of pages	5
Journal	Electronic Journal of Linear Algebra
Volume	16
DOIs	https://doi.org/10.13001/1081-3810.1183
State	Published - 30 Jan 2007

Keywords

Cayley formula
Laplacian
Number of leaves
Random spanning trees
Spectrum

ASJC Scopus subject areas

Algebra and Number Theory

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10.13001/1081-3810.1183

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abstract = "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 ≤iv λi+1, where λi is the ith smallest eigenvalues of the Laplacian of the original graph and λiv 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, if λ2 > αn.",

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AB - 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 ≤iv λi+1, where λi is the ith smallest eigenvalues of the Laplacian of the original graph and λiv 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, if λ2 > αn.

KW - Cayley formula

KW - Laplacian

KW - Number of leaves

KW - Random spanning trees

KW - Spectrum

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ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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