Abstract
Matrix representations for graphs play an important role in combinatorics. In this paper, we investigate four matrix representations for graphs and carry out an characteristic polynomial analysis upon them. The first two graph matrices are the adjacency matrix and Laplacian matrix. These two matrices can be obtained straightforwardly from graphs. The second two matrix representations, which are newly introduced [9][3], are closely related with the Ihara zeta function and the discrete time quantum walk. They have a similar form and are established from a transformed graph, i.e. the oriented line graph of the original graph. We make use of the characteristic polynomial coefficients of the four matrices to characterize graphs and construct pattern spaces with a fixed dimensionality. Experimental results indicate that the two matrices in the transformed domain perform better than the two matrices in the original graph domain whereas the matrix associated with the Ihara zeta function is more efficient and effective than the matrix associated with the discrete time quantum walk and the remaining methods.
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Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: Quantum walks on graphs. In: Proceedings of ACM Theory of Computing (2001)
Bai, X., Hancock, E.R., Wilson, R.C.: Graph characteristics from the heat kernel trace. In: Pattern Recognition (2009) (to appear)
Emms, D., Severini, S., Wilson, R.C., Hancock, E.R.: Coined quantum walks lift the cospectrality of graphs and trees. In: Proceedings of SSPR (2008)
Ihara, Y.: Discrete subgroups of pl(2, k ϕ ). In: Proceeding Symposium of Pure Mathematics, pp. 272–278 (1965)
Ihara, Y.: On discrete subgroups of the two by two projective linear group over p-adic fields. Journal of Mathematics Society Japan 18, 219–235 (1966)
Kotani, M., Sunada, T.: Zeta functions of finite graphs. Journal of Mathematics University of Tokyo 7(1), 7–25 (2000)
Luo, B., Wilson, R.C., Hancock, E.R.: Spectral embedding of graphs. Pattern Recognition 36(10), 2213–2223 (2003)
Ren, P., Wilson, R.C., Hancock, E.R.: Graph characteristics from the ihara zeta function. In: Proceedings of SSPR (2008)
Ren, P., Wilson, R.C., Hancock, E.R.: Pattern vectors from the ihara zeta function. In: Proceedings of The 19th International Conference of Pattern Recognition (2008)
Wilson, R.C., Luo, B., Hancock, E.R.: Pattern vectors from algebraic graph theory. IEEE Transactions on Pattern Analysis and Machine Intelligence 27(7), 1112–1124 (2005)
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© 2009 Springer-Verlag Berlin Heidelberg
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Ren, P., Wilson, R.C., Hancock, E.R. (2009). Characteristic Polynomial Analysis on Matrix Representations of Graphs. In: Torsello, A., Escolano, F., Brun, L. (eds) Graph-Based Representations in Pattern Recognition. GbRPR 2009. Lecture Notes in Computer Science, vol 5534. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02124-4_25
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DOI: https://doi.org/10.1007/978-3-642-02124-4_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02123-7
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