A study of graph spectra for comparing graphs and trees

The spectrum of a graph has been widely used in graph theory to characterise the properties of a graph and extract information from its structure. It has also been employed as a graph representation for pattern matching since it is invariant to the labelling of the graph. There are, however, a number of potential drawbacks in using the spectrum as a representation of a graph. Firstly, more than one graph may share the same spectrum. It is well known, for example, that very few trees can be uniquely specified by their spectrum. Secondly, the spectrum may change dramatically with a small change structure. There are a wide variety of graph matrix representations from which the spectrum can be extracted. Among these are the adjacency matrix, combinatorial Laplacian, normalised Laplacian and unsigned Laplacian. Spectra can also be derived from the heat kernel matrix and path length distribution matrix. The choice of matrix representation clearly has a large effect on the suitability of spectrum in a number of pattern recognition tasks. In this paper we investigate the performance of the spectra as a graph representation in a variety of situations. Firstly, we investigate the cospectrality of the various matrix representations over large graph and tree sets, extending the work of previous authors. We then show that the Euclidean distance between spectra tracks the edit distance between graphs over a wide range of edit costs, and we analyse the accuracy of this relationship. We then use the spectra to both cluster and classify the graphs and demonstrate the effect of the graph matrix formulation on error rates. These results are produced using both synthetic graphs and trees and graphs derived from shape and image data.

[1]  Edwin R. Hancock,et al.  Pattern spaces from graph polynomials , 2003, 12th International Conference on Image Analysis and Processing, 2003.Proceedings..

[2]  Edwin R. Hancock,et al.  Graph manifolds from spectral polynomials , 2004, Proceedings of the 17th International Conference on Pattern Recognition, 2004. ICPR 2004..

[3]  M. Fatih Demirci,et al.  Indexing through laplacian spectra , 2008, Comput. Vis. Image Underst..

[4]  King-Sun Fu,et al.  A distance measure between attributed relational graphs for pattern recognition , 1983, IEEE Transactions on Systems, Man, and Cybernetics.

[5]  Ali Shokoufandeh,et al.  Shock Graphs and Shape Matching , 1998, International Journal of Computer Vision.

[6]  Martin A. Fischler,et al.  The Representation and Matching of Pictorial Structures , 1973, IEEE Transactions on Computers.

[7]  Shinji Umeyama,et al.  An Eigendecomposition Approach to Weighted Graph Matching Problems , 1988, IEEE Trans. Pattern Anal. Mach. Intell..

[8]  Fan Chung,et al.  Spectral Graph Theory , 1996 .

[9]  Edwin R. Hancock,et al.  Clustering shapes using heat content invariants , 2005, IEEE International Conference on Image Processing 2005.

[10]  Richard C. Wilson,et al.  Contour Segments from Spline Interpolation , 2004, SSPR/SPR.

[11]  Edwin R. Hancock,et al.  Heat Kernels, Manifolds and Graph Embedding , 2004, SSPR/SPR.

[12]  Ali Shokoufandeh,et al.  Retrieving Articulated 3-D Models Using Medial Surfaces and Their Graph Spectra , 2005, EMMCVPR.

[13]  Willem H. Haemers,et al.  Enumeration of cospectral graphs , 2004, Eur. J. Comb..

[14]  Steven Gold,et al.  A Graduated Assignment Algorithm for Graph Matching , 1996, IEEE Trans. Pattern Anal. Mach. Intell..

[15]  Horst Bunke,et al.  Error Correcting Graph Matching: On the Influence of the Underlying Cost Function , 1999, IEEE Trans. Pattern Anal. Mach. Intell..

[16]  Kim L. Boyer,et al.  Perceptual organization in computer vision: a review and a proposal for a classificatory structure , 1993, IEEE Trans. Syst. Man Cybern..

[17]  F. Harary.  New directions in the theory of graphs , 1973 .

[18]  Terry Caelli,et al.  Inexact Multisubgraph Matching Using Graph Eigenspace and Clustering Models , 2002, SSPR/SPR.

[19]  Frank Ruskey,et al.  The advantages of forward thinking in generating rooted and free trees , 1999, SODA '99.

[20]  W. Haemers,et al.  Which graphs are determined by their spectrum , 2003 .

[21]  Edwin R. Hancock,et al.  Structural Matching by Discrete Relaxation , 1997, IEEE Trans. Pattern Anal. Mach. Intell..

[22]  Horst Bunke,et al.  On a relation between graph edit distance and maximum common subgraph , 1997, Pattern Recognit. Lett..

[23]  Barend J. van Wyk,et al.  A POCS-Based Graph Matching Algorithm , 2004, IEEE Trans. Pattern Anal. Mach. Intell..

[24]  Edwin R. Hancock,et al.  Learning shape-classes using a mixture of tree-unions , 2006, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[25]  Edwin R. Hancock,et al.  A Riemannian approach to graph embedding , 2007, Pattern Recognit..

[26]  Jitendra Malik,et al.  Normalized cuts and image segmentation , 1997, Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[27]  Sergey Gubin,et al.  On Subgraph Isomorphism , 2008, ArXiv.

[28]  Ali Shokoufandeh,et al.  Indexing using a spectral encoding of topological structure , 1999, Proceedings. 1999 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (Cat. No PR00149).

[29]  B. McKay,et al.  Constructing cospectral graphs , 1982 .

[30]  Sameer A. Nene,et al.  Columbia Object Image Library (COIL100) , 1996 .

[31]  Harry G. Barrow,et al.  Subgraph Isomorphism, Matching Relational Structures and Maximal Cliques , 1976, Inf. Process. Lett..

[32]  Michael E. Taylor,et al.  Differential Geometry I , 1994 .