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Abstract

This paper explores how to extend the spectral analysis of graphs to the case where the nodes and edges are attributed. To do this we introduce a complex Hermitian variant of the Laplacian matrix. Our spectral representation is based on the eigendecomposition of the resulting Hermitian property matrix. The eigenvalues of the matrix are real while the eigenvectors are complex. We show how to use symmetric polynomials to construct permutation invariants from the elements of the resulting complex spectral matrix. We construct pattern vectors from the resulting invariants, and use them to embed the graphs in a low dimensional pattern space using a number of well-known techniques including principal components analysis, linear discriminant analysis and multidimensional scaling.

Keywords

Linear Discriminant Analysis Adjacency Matrix Multidimensional Scaling Edge Weight Hermitian Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Chung, F.R.K.: Spectral Graph Theory. American Mathmatical Society Ed., CBMS series 92 (1997)Google Scholar
  2. 2.
    Umeyama, S.: An eigen decomposition approach to weighted graph matching problems. IEEE Transactions on Pattern Analysis and Machine Intelligence 10, 695–703 (1988)zbMATHCrossRefGoogle Scholar
  3. 3.
    Luo, B., Wilson, R.C., Hancock, E.R.: Spectral Embedding of Graphs. Pattern Recognition 36(10), 2213–2230 (2003)zbMATHCrossRefGoogle Scholar
  4. 4.
    Mohar, B.: Laplace Eigenvalues of Graphs - A survey. Discrete Mathematics 109, 171–183 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Biggs, N.: Algebraic Graph Theory, CUPGoogle Scholar
  6. 6.
    Huet, B., Hancock, E.R.: Line Pattern Retrieval Using Relational Histograms. IEEE Transactions on Pattern Analysis and Machine Intelligence 21, 1363–1370 (1999)CrossRefGoogle Scholar
  7. 7.
    Siddiqi, K., Shokoufandeh, A., Dickinson, S.J., Zucker, S.W.: Shock Graphs and Shape Matching. International Journal of Computer Vision 35(1), 13–32 (1999)CrossRefGoogle Scholar
  8. 8.
    Shi, J., Malik, J.: Normalized Cuts and Image Segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 22(8), 888–905 (2000)CrossRefGoogle Scholar
  9. 9.
    Torsello, A., Hancock, E.R.: A skeletal measure of 2D shape similarity. In: Arcelli, C., Cordella, L.P., Sanniti di Baja, G. (eds.) IWVF 2001. LNCS, vol. 2059, pp. 594–605. Springer, Heidelberg (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Richard C. Wilson
    • 1
  • Edwin R. Hancock
    • 1
  1. 1.Department of Computer ScienceUniversity of YorkYorkUK

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