Journal of Algebraic Combinatorics

, Volume 24, Issue 2, pp 195–207 | Cite as

Smith’s Theorem and a characterization of the 6-cube as distance-transitive graph

Article

Abstract

A generic distance-regular graph is primitive of diameter at least two and valency at least three. We give a version of Derek Smith's famous theorem for reducing the classification of distance-regular graphs to that of primitive graphs. There are twelve cases—the generic case, four canonical imprimitive cases that reduce to the generic case, and seven exceptional cases. All distance-transitive graphs were previously known in six of the seven exceptional cases. We prove that the 6-cube is the only distance-transitive graph coming under the remaining exceptional case.

Keywords

Imprimitive distance-transitive graph Imprimitive distance-regular graph 

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Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Department of Mathematical SciencesKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia
  2. 2.Department of MathematicsMichigan State UniversityEast LansingUSA

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