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1996 Fiscal Year Final Research Report Summary

Structure of Distance-Regular Graphs

Research Project

Project/Area Number 06640075
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionInternational Christian University

Principal Investigator

SUZUKI Hiroshi  International Christian University, Mathematics, Professor, 教養学部, 教授 (10135767)

Project Period (FY) 1994 – 1996
Keywordsalgebraic combinatorics / distance-regular graph / association scheme / finite geometry / algebraic graph theory / representation of finite groups
Research Abstract

(1) Great progress in the study of the incidence graph of a distance-regular graph was made. The concept of distance-semi-regular graphs was defined in [1]. The class contains distance-biregular graphs and the incidence graphs of regular near polygons. By a characterization of distance-regular graphs whose incidence graphs become distance-semi-regular graphs, the study of distance-regular graphs of order (s, t) with s>t became very simple. For example, the study of distance-regular graphs of order (s, 2) was done by very involved combinatorial arguments previously, but it was replaced by a very simple parameter analysis.
(2) Study of the existence condition of a large subgraphs is a most interesting field in distance-regular graphs, as it sometimes links with the bound of the geometrical girth. A.Hiraki and C-W.Weng showed independently the existence of strongly closed subgraphs of distance-regular graphs. In [2], the author could combine the technique to show the existence of a series … More of strongly closed subgraphs when the geometric girth of the graph is five, which is the case C-W.Weng could not hadle. It is a hope to study geometry of distance-regular graphs by the use of subgraphs of them.
(3) To study the representation theory of distance-regular graphs, it is absolutely necessary to study Q-polynomial association schemes. But until recently, there were no clue to study parametrical conditions of Q-polynomial association schemes as most of the counter part of distance-regular graphs were proved by combinatorial arguments. In [3], hinted by a work of G.Dickie, the author could prove several vanishing conditions of parameters to give a characterization of imprimitive Q-polynomial association schemes, which was one of the open problems for a long time. He also could determine association schemes with multiple Q-polynomial structures [4]. He hopes that the study of Q-polynomial association schemes sheds light to the representation theory of association schemes and distance-regular graphs.
(4) Character product of finite groups is very much related to the representation theory of association schemes. In the investigation of Q-polynomial association schemes in the class of group schemes, the author could classify finite groups having irreducible characters chi, psi such that chi^2=1+mpsi for some integer m in [5]. By this, Q-polynomial association schemes which come from group association schemes were classified under a condition that a^**_=0. He hopes that the study of representation theory of association schemes can also be a stimulation to other field of mathematics. Less

  • Research Products

    (9 results)

All Other

All Publications (9 results)

  • [Publications] H.Suzuki: "Distance Semi-Regular Graphs" Algebra Colloquium. 2. 315-328 (1995)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Suzuki: "Strongly Closed Subgraphs of a Distance-Regular Graph of Geometric Girth Five" Kyushu Journal of Mathematics. 50. 371-384 (1996)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Suzuki: "Imprimitive Q-polynomial Association Schemes" Journal of Algebraic Combinatorics. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H.Suzuki: "Association Schemes with Multiple Q-polynomial Structures" Journal of Algebraic Combinatorics. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] "Distance Semi-Regular Graphs" Algebraic Colloquium. 2. 315-328 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] "Strongly Closed Subgraphs of a Distance-Regular Graph of Geometric Girth Five" Kyushu Journal of Mathematics. 50. 371-384 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] "Imprimitive Q-polynomial Association Schemes" Journal of Algebraic Combinatorics. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] "Association Schemes with Multiple Q-polynomial Structures" Journal of Algebraic Combinatorics. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] "Character Products, and Balanced Sets Associated with Group Schemes" (preprint).

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 1999-03-09  

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