DMCA
Distance-Regular Graphs Having an Eigenvalue of Small Multiplicity
BibTeX
@MISC{Martin_distance-regulargraphs,
author = {William J. Martin and Rupert Ruopeng Zhu},
title = {Distance-Regular Graphs Having an Eigenvalue of Small Multiplicity},
year = {}
}
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Abstract
This paper is a contribution to the classification of distance-regular graphs by eigenvalue multiplicity. Here we determine the distance-regular graphs with an eigenvalue of multiplicity five, six, or seven. The approach uses the representation theory of distance-regular graphs and, in particular, results of Godsil [12] and Terwilliger [19], to allow computer generation of all feasible intesection arrays. The classification results are then proved by appealing to known theory. Based on arguments of Godsil and Terwilliger and on results in this paper, we now have the diameter bound d 3m \Gamma 8 for m 5. 1 Introduction It is well-known that a distance-regular graph having an eigenvalue of multiplicity two is either a cycle or a complete multipartite graph with three equal parts. The distance-regular graphs with an eigenvalue of multiplicity three are precisely the one-skeletons of the five Platonic solids and all complete multipartite graphs with four equal parts. In [22, 23], Zhu ...
