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International Journal of Mathematics and Mathematical Sciences
Volume 27, Issue 2, Pages 111-123
http://dx.doi.org/10.1155/S0161171201002071

The number of edges on generalizations of Paley graphs

Department of Mathematics, California Polytechnic University, San Luis Obispo 93407, CA, USA

Received 21 August 1997

Copyright © 2001 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Evans, Pulham, and Sheenan computed the number of complete 4-subgraphs of Paley graphs by counting the number of edges of the subgraph containing only those nodes x for which x and x1 are quadratic residues. Here we obtain formulae for the number of edges of generalizations of these subgraphs using Gaussian hypergeometric series and elliptic curves. Such formulae are simple in several infinite families, including those studied by Evans, Pulham, and Sheenan.