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Spectrum of a homogeneous graph. (English) Zbl 1152.05348
Summary: We describe the spectrum of the Laplacian for a homogeneous graph on which a discrete group acts. This follows from a more general result which describes the spectrum of a convolution operator on a homogeneous space of a locally compact group. We also prove a version of Harnack inequality for a Schrödinger operator on an invariant homogeneous graph.
##### MSC:
 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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##### References:
 [1] Baggett, L.; Packer, J., The primitive ideal space of two-step nilpotent group $$C^\ast$$-algebras, J. funct. anal., 124, 389-426, (1994) · Zbl 0820.22004 [2] Chu, C.-H., Harmonic function spaces on groups, J. London math. soc., 70, 182-198, (2004) · Zbl 1054.43002 [3] Chu, C.-H., Matrix convolution operators on groups, Lecture notes in math., vol. 1956, (2008), Springer-Verlag Heidelberg [4] Chu, C.-H.; Leung, C.-W., The convolution equation of Choquet and deny on [IN]-groups, Integral equations operator theory, 40, 391-402, (2001) · Zbl 1033.43001 [5] Chung, F.R.K., Spectral graph theory, CBMS leture notes, (1997), Amer. Math. Soc. Providence, RI · Zbl 0872.05052 [6] Chung, F.R.K.; Sternberg, S., Laplacian and vibrational spectra for homogeneous graphs, J. graph theory, 16, 605-627, (1992) · Zbl 0768.05049 [7] Chung, F.R.K.; Yau, S.-T., A Harnack inequality for homogeneous graphs and subgraphs, Comm. anal. geom., 2, 628-639, (1994) [8] Chung, F.R.K.; Yau, S.-T., A Harnack inequality for Dirichlet eigenvalues, J. graph theory, 34, 247-257, (2000) · Zbl 0953.05045 [9] Dixmier, J., LES $$C^\ast$$-algèbres et leur représentations, (1969), Gauthier-Villar Paris · Zbl 0174.18601 [10] Dodziuk, J.; Kendall, W.S., Combinatorial Laplacians and isoperimetric inequality, (), 68-75 · Zbl 0619.05005 [11] Folland, E.B., A course in abstract harmonic analysis, (1995), CRC Press Boca Raton · Zbl 0857.43001 [12] Mohar, B., Isoperimetric inequalities, growth, and the spectrum of graphs, Linear algebra appl., 103, 119-131, (1988) · Zbl 0658.05055 [13] Pedersen, G.K., $$C^\ast$$-algebras and their automorphism groups, (1979), Academic Press London [14] Tan, J., On Cheeger inequalities of a graph, Discrete math., 269, 315-323, (2003) · Zbl 1021.05070 [15] Tandra, H.; Moran, W., Characters of the discrete Heisenberg group and of its completion, Math. proc. Cambridge philos. soc., 136, 525-539, (2004) · Zbl 1052.43008 [16] Urakawa, H., The Cheeger constant, the heat kernel, and the Green kernel of an infinite graph, Monatsh. math., 138, 225-237, (2003) · Zbl 1053.05084
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