1996 Fiscal Year Final Research Report Summary
Structure of Distance-Regular Graphs
| Project/Area Number |
06640075
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | International Christian University |
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Principal Investigator |
SUZUKI Hiroshi International Christian University, Mathematics, Professor, 教養学部, 教授 (10135767)
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| Project Period (FY) |
1994 – 1996
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| Keywords | algebraic combinatorics / distance-regular graph / association scheme / finite geometry / algebraic graph theory / representation of finite groups |
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| 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
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