The Laplacian of a Graph
The Laplacian is another important matrix associated with a graph, and the Laplacian spectrum is the spectrum of this matrix. We will consider the relationship between structural properties of a graph and the Laplacian spectrum, in a similar fashion to the spectral graph theory of previous chapters. We will meet Kirchhoff’s expression for the number of spanning trees of a graph as the determinant of the matrix we get by deleting a row and column from the Laplacian. This is one of the oldest results in algebraic graph theory. We will also see how the Laplacian can be used in a number of ways to provide interesting geometric representations of a graph. This is related to work on the Colin de Verdiere number of a graph, which is one of the most important recent developments in graph theory.
KeywordsSpan Tree Adjacency Matrix Connected Graph Regular Graph Hamilton Cycle
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- H. van der Holst, L. Lovász, and A. Schrijver, The Colin de Verdière graph parameter, in Graph theory and combinatorial biology (Balatonlelle, 1996), Janos Bolyai Math. Soc, Budapest, 1999, 29–85.Google Scholar