Non-Ramanujancy of Euclidean graphs of order 2 r

Abstract

Graphs are attached to the n-dimensional space Z 2 r n where Z 2 r is the ring with 2 r elements using an analogue of Euclidean distance. The graphs are shown to be non-Ramanujan for r=4. Comparisons are made with Euclidean graphs attached to Z p r n for p an odd prime. The percentage of non-zero eigenvalues of the adjacency operator attached to these finite Euclidean graphs is shown to tend to zero as n tends to infinity.