Budget Amount *help 
¥1,800,000 (Direct Cost : ¥1,800,000)
Fiscal Year 1996 : ¥600,000 (Direct Cost : ¥600,000)
Fiscal Year 1995 : ¥600,000 (Direct Cost : ¥600,000)
Fiscal Year 1994 : ¥600,000 (Direct Cost : ¥600,000)

Research Abstract 
(1) Great progress in the study of the incidence graph of a distanceregular graph was made. The concept of distancesemiregular graphs was defined in [1]. The class contains distancebiregular graphs and the incidence graphs of regular near polygons. By a characterization of distanceregular graphs whose incidence graphs become distancesemiregular graphs, the study of distanceregular graphs of order (s, t) with s>t became very simple. For example, the study of distanceregular 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 distanceregular graphs, as it sometimes links with the bound of the geometrical girth. A.Hiraki and CW.Weng showed independently the existence of strongly closed subgraphs of distanceregular graphs. In [2], the author could combine the technique to show the existence of a series
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of strongly closed subgraphs when the geometric girth of the graph is five, which is the case CW.Weng could not hadle. It is a hope to study geometry of distanceregular graphs by the use of subgraphs of them. (3) To study the representation theory of distanceregular graphs, it is absolutely necessary to study Qpolynomial association schemes. But until recently, there were no clue to study parametrical conditions of Qpolynomial association schemes as most of the counter part of distanceregular 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 Qpolynomial association schemes, which was one of the open problems for a long time. He also could determine association schemes with multiple Qpolynomial structures [4]. He hopes that the study of Qpolynomial association schemes sheds light to the representation theory of association schemes and distanceregular graphs. (4) Character product of finite groups is very much related to the representation theory of association schemes. In the investigation of Qpolynomial 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, Qpolynomial 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
