Eigenvalues of the Laplacian and Their Relationship to the Connectedness of a Graph
نویسنده
چکیده
This paper develops the necessary tools to understand the relationship between eigenvalues of the Laplacian matrix of a graph and the connectedness of the graph. First we prove that a graph has k connected components if and only if the algebraic multiplicity of eigenvalue 0 for the graph’s Laplacian matrix is k. We then prove Cheeger’s inequality (for dregular graphs) which bounds the number of edges between the two subgraphs of G that are the least connected to one another using the second smallest eigenvalue of the Laplacian of G.
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