Equiangular lines and covers of the complete graph

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• 作者：G. Coutinho ; ^{gcoutinho@uwaterloo.ca" class="auth_mail" title="E-mail the corresponding author} ; C. Godsil ^{cgodsil@uwaterloo.ca" class="auth_mail" title="E-mail the corresponding author} ; H. Shirazi ^{hamedshirazi@alumni.uwaterloo.ca" class="auth_mail" title="E-mail the corresponding author} ; H. Zhan ^{h3zhan@uwaterloo.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：05E30 ; 15B33 ; 42C15
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 January 2016
• 年：2016
• 卷：488
• 期：Complete
• 页码：264-283
• 全文大小：393 K

文摘

The relation between equiangular sets of lines in the real space and distance-regular double covers of the complete graph is well known and studied since the work of Seidel and others in the 70s. The main topic of this paper is to continue the study on how complex equiangular lines relate to distance-regular covers of the complete graph with larger index. Given a set of equiangular lines meeting the relative (or Welch) bound, we show that if the entries of the corresponding Gram matrix are prime roots of unity, then these lines can be used to construct an antipodal distance-regular graph of diameter three. We also study in detail how the absolute (or Gerzon) bound for a set of equiangular lines can be used to derive bounds of the parameters of abelian distance-regular covers of the complete graph.