Algebraic Graph Theory pp 279-306 | Cite as

# The Laplacian of a Graph

## 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

Span Tree Adjacency Matrix Connected Graph Regular Graph Hamilton Cycle## Preview

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## References

- [1]W. N. Anderson Jr. and T. D. Morley,
*Eigenvalues of the Laplacian of a graph*, Linear and Multilinear Algebra, 18 (1985), 141–145.MathSciNetMATHCrossRefGoogle Scholar - [2]M. Fiedler,
*Algebraic connectivity of graphs*, Czechoslovak Math. J., 23(98) (1973), 298–305.MathSciNetGoogle Scholar - [3]—,
*A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory*, Czechoslovak Math. J., 25(100) (1975), 619–633.MathSciNetGoogle Scholar - [4]L. Lovász,
*Combinatorial Problems and Exercises*, North-Holland Publishing Co., Amsterdam, second edition, 1993.MATHGoogle Scholar - [5]L. Lovász and A. Schrijver,
*On the null space of a Colin de Verdiere matrix*, Ann. Inst. Fourier (Grenoble), 49 (1999), 1017–1026.MathSciNetMATHCrossRefGoogle Scholar - [6]B. Mohar,
*Isoperimetric numbers of graphs*, J. Combin. Theory Ser. B, 47 (1989), 274–291.MathSciNetMATHCrossRefGoogle Scholar - [7]—,
*Eigenvalues, diameter, and mean distance in graphs*, Graphs Combin., 7 (1991), 53–64.MathSciNetMATHCrossRefGoogle Scholar - [8]T. Pisanskl and J. Shawe-Taylor,
*Characterising graph drawing with eigenvectors*, J. Chem. Inf. Comput. Sci, 40 (2000), 567–571.CrossRefGoogle Scholar - [9]W. T. Tutte,
*How to draw a graph*, Proc. London Math. Soc. (3), 13 (1963), 743–767.MathSciNetMATHCrossRefGoogle Scholar - [10]E. R. van Dam and W. H. Haemers,
*Graphs with constant μ and*\( \bar \mu \), Discrete Math., 182 (1998), 293–307.MathSciNetMATHCrossRefGoogle Scholar - [11]J. van den Heuvel,
*Hamilton cycles and eigenvalues of graphs*, Linear Algebra Appl., 226/228 (1995), 723–730.MathSciNetCrossRefGoogle Scholar - [12]H. van der Holst,
*A short proof of the planarity characterization of Colin de Verdiere*, J. Combin. Theory Ser. B, 65 (1995), 269–272.MathSciNetMATHCrossRefGoogle Scholar - [13]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