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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
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Abstract: We consider graphs attached to $(\mathbb {Z}/q\mathbb {Z})^n$, where $q=p^r$, for an odd prime $p$, 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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  • 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 ($\mathop{\rm mod}p^n),n\geq 2$, Bull. London Math. Society, 5 (1973), 325-327. MR 48:6020
  • 8. Hans Salié, Über die Kloostermanschen Summen $S(u,v;q)$, 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

A. Terras

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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