The Laplacian of a Graph

  • Chris Godsil
  • Gordon Royle
Part of the Graduate Texts in Mathematics book series (GTM, volume 207)

Abstract

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.

Keywords

Hull Posit Eter Bedding Elon 

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Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Chris Godsil
    • 1
  • Gordon Royle
    • 2
  1. 1.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada
  2. 2.Department of Computer ScienceUniversity of Western AustraliaNedlandsAustralia

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