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Finite Euclidean graphs over rings
Authors:
A. Medrano, P. Myers, H. M. Stark and A. Terras
Journal:
Proc. Amer. Math. Soc. 126 (1998), 701-710
MSC (1991):
Primary 11T23; Secondary 05C25
MathSciNet review:
1443395
Full-text PDF Free Access
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Abstract: We consider graphs attached to  , where  , for an odd prime  , using an analogue of the Euclidean distance. These graphs are shown to be mostly non-Ramanujan, in contrast to the case of Euclidean graphs over finite fields.
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Angel, C.
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- 1.
- Jeff Angel, Bernadette Shook, Audrey Terras, and Cindy Trimble, Graph Spectra for Finite Upper Half Planes over rings, Linear Algebra Appl., 226-228 (1995), 423-457. MR 96g:05094
- 2.
- Wen-Ching Winnie Li, A survey of Ramanujan graphs, in R. Pellikaan, M. Perret and S.G. Vladut (Eds.), Arithmetic, Geometry and Coding Theory, Proc. Conf. at C.I.R.M., Luminy, June 28-July 2, 1993, de Gruyter, Berlin, 1996. CMP 96:14
- 3.
- Alexander Lubotzky, Discrete Groups, Expanding Graphs and Invariant Measures, Birkhäuser, Basel, 1994. MR 96g:22018
- 4.
- Alexander Lubotzky, Ralph Phillips, and Peter Sarnak, Ramanujan graphs, Combinatorica, 8 (1988), 261-277. MR 89m:05099
- 5.
- Archie Medrano, Perla Myers, Harold M. Stark, and Audrey Terras, Finite analogues of Euclidean space, J. Comput. Appl. Math. 68 (1996), 221-238. CMP 97:04
- 6.
- Perla Myers, Ph.D. Thesis, U.C.S.D., 1995.
- 7.
- R.W.K. Odoni, On Gauss sums (
 , Bull. London Math. Society, 5 (1973), 325-327. MR 48:6020
- 8.
- Hans Salié, Über die Kloostermanschen Summen , Math Z., 34 (1931), 91-109.
- 9.
- Peter Sarnak, Some Applications of Modular Forms, Cambridge U. Press, Cambridge, 1990. MR 92k:11045
- 10.
- Harold M. Stark and Audrey Terras, Zeta functions of finite graphs and coverings, Advances in Math., 121 (1996), 124-165. CMP 96:15
- 11.
- Audrey Terras, Survey of spectra of Laplacians on finite symmetric spaces, Experimental Math., 5 (1996), 15-32. CMP 97:02
- 12.
- Audrey Terras, Harmonic Analysis on Symmetric Spaces and Applications, I,II, Springer-Verlag, New York, 1985, 1988. MR 87f:22010; MR 89k:22017
- 13.
- Albert L. Whiteman, A note on Kloosterman sums, B.A.M.S., 51 (1945), 373-377. MR 6:259f
- 14.
- Kenneth S. Williams, Note on the Kloosterman sum, Proc. A.M.S., 30 (1971), 61-62. MR 44:2719
- 15.
- Kenneth S. Williams, The Kloosterman sum revisited, Canadian Math. Society Bull.,16 (1973), 363-365. MR 48:6021
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Additional Information
A. Medrano
Affiliation:
Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0112
P. Myers
Affiliation:
Department of Mathematics, University of California, Santa Cruz, California 95064
Email:
pmyers@cats.ucsc.edu
A. Terras
Email:
aterras@ucsd.edu
DOI:
http://dx.doi.org/10.1090/S0002-9939-98-04294-4
PII:
S 0002-9939(98)04294-4
Keywords:
Euclidean graph,
Ramanujan graph,
Kloosterman sums over rings
Received by editor(s):
September 11, 1996
Communicated by:
Dennis A. Hejhal
Article copyright:
© Copyright 1998 American Mathematical Society
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