Completely regular clique graphs, II

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• 作者：Hiroshi Suzuki
• 关键词：Distance ; regular graph ; Association scheme ; Subconstituent algebra ; Terwilliger algebra ; Completely regular code ; Distance ; semiregular graph
• 刊名：Journal of Algebraic Combinatorics
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：43
• 期：2
• 页码：417-445
• 全文大小：579 KB
• 参考文献：1.Bang, S., Hiraki, A., Koolen, J.H.: Delsarte clique graphs. Eur. J. Combin. 28, 501–516 (2007)CrossRef MathSciNet MATH
  2.Bang, S., Hiraki, A., Koolen, J.H.: Delsarte set graphs with small \(c_2\) . Graphs Combin. 26, 147–162 (2010)CrossRef MathSciNet MATH
  3.Bang, S., Koolen, J.H.: On distance-regular graphs with smallest eigenvalue at least \(-m\) . J. Combin. Theory Ser. B 100, 573–584 (2010)CrossRef MathSciNet MATH
  4.Bannai, E., Ito, T.: Algebraic Combinatorics I. Benjamin-Cummings, California (1984)MATH
  5.Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs. Springer, Berlin, Heidelberg (1989)CrossRef MATH
  6.Cámara, M., Dalfó, C., Delorme, C., Fiol, M.A., Suzuki, H.: Edge-distance-regular graphs are distance-regular. J. Combin. Theory Ser. A 120, 1057–1067 (2013)CrossRef MathSciNet MATH
  7.Delorme, C.: Distance biregular bipartite graphs. Eur. J. Combin. 15, 223–238 (1994)CrossRef MathSciNet MATH
  8.Fiol, M.A.: Pseudo-distance-regularized graphs are distance-regular or distance biregular. Linear Algebra Appl. 437, 2973–2977 (2012)CrossRef MathSciNet MATH
  9.Godsil, C.D., Shawe-Taylor, J.: Distance-regularised graphs are distance-regular or distance-biregular. J. Combin. Theory Ser. B 43, 14–24 (1987)CrossRef MathSciNet MATH
  10.Leonard, D.A.: Parameters of association schemes that are both \(P\) - and \(Q\) -polynomial. J. Combin. Theory Ser. A 36, 355–363 (1984)CrossRef MathSciNet MATH
  11.Neumaier, A.: Completely regular codes. Discrete Math. 106(107), 353–360 (1992)CrossRef MathSciNet
  12.Suzuki, H.: Completely regular clique graphs. J. Algebr. Combin. 40, 233–244 (2014)CrossRef MATH
  13.Suzuki, H.: The geometric girth of a distance-regular graph having certain thin irreducible modules for the Terwilliger algebra. Eur. J. Combin. 27, 235–254 (2006)CrossRef MATH
  14.Suzuki, H.: The Terwilliger algebra associated with a set of vertices in a distance-regular graph. J. Algebr. Combin. 22, 5–38 (2005)CrossRef MATH
  15.Suzuki, H.: Distance-semiregular graphs. Algebra Colloq. 2, 315–328 (1995)MathSciNet MATH
  16.Terwilliger, P.: Algebraic graph theory. Hand-written note of a series of lectures given in 1993, rewritten and added comments by H. Suzuki. http://​subsite.​icu.​ac.​jp/​people/​hsuzuki/​lecturenote/​
  17.Terwilliger, P.: The subconstituent algebra of an association scheme, Part I. J. Algebr. Combin. 1, 363–388 (1992)CrossRef MathSciNet MATH
  18.Terwilliger, P.: The subconstituent algebra of an association scheme, Part II. J. Algebr. Combin. 2, 73–103 (1993)CrossRef MathSciNet MATH
  19.Terwilliger, P.: The subconstituent algebra of an association scheme, Part III. J. Algebr. Combin. 2, 177–210 (1993)CrossRef MathSciNet MATH
  20.Terwilliger, P.: The subconstituent algebra of a distance-regular graph; thin modules with endpoint one. Linear Algebra Appl. 356, 157–187 (2002)CrossRef MathSciNet MATH
  
• 作者单位：Hiroshi Suzuki (1)
  
  1. International Christian University, Mitaka, Tokyo, 181-8585, Japan
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Combinatorics
  Convex and Discrete Geometry
  Order, Lattices and Ordered Algebraic Structures
  Computer Science, general
  Group Theory and Generalizations
  
• 出版者：Springer U.S.
• ISSN：1572-9192

文摘

Let \(\varGamma = (X,R)\) be a connected graph. Then \(\varGamma \) is said to be a completely regular clique graph of parameters (s, c) with \(s\ge 1\) and \(c\ge 1\), if there is a collection \({\mathcal {C}}\) of completely regular cliques of size \(s+1\) such that every edge is contained in exactly c members of \({\mathcal {C}}\). In the previous paper (Suzuki in J Algebr Combin 40:233–244, 2014), we showed, among other things, that a completely regular clique graph is distance-regular if and only if it is a bipartite half of a certain distance-semiregular graph. In this paper, we show that a completely regular clique graph with respect to \({\mathcal {C}}\) is distance-regular if and only if every \({\mathcal {T}}(C)\)-module of endpoint zero is thin for all \(C\in {\mathcal {C}}\). We also discuss the relation between a \({\mathcal {T}}(C)\)-module of endpoint 0 and a \({\mathcal {T}}(x)\)-module of endpoint 1 and study examples of completely regular clique graphs. Keywords Distance-regular graph Association scheme Subconstituent algebra Terwilliger algebra Completely regular code Distance-semiregular graph