zbMATH — the first resource for mathematics

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.
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.