Doubly Regular Asymmetric Digraphs And Non-symmetric Association Schemes

Association scheme is a key topic in algebraic combinatorics. The graph and associa-tion scheme share some common properties. The graph deals with points and edges, and the association scheme deals with treatments. We regard points in association scheme as points in graph, and regard relations in association scheme as edgs in graph. Some association schemes can be expressed by the graphs. Doubly regular asymmetric digraph is a special kind of graph. Ionin and Kharaghani constructed doubly regular asymmetric digraph by using the symmetric design. We find that the Bose-Mesner algebra of doubly regular asymmetric digraph is similar to the Bose-Mesner of non-symmetric association scheme. Motivated by this, we give the sufficient condition for doubly regular asymmet-ric digraph to be a association scheme. Moreover, we illustrate the close ties between association scheme and symmetric design.The paper is composed of three chapters and organized as follows:In Chapter1, we introduce the definitions and properties of doubly regular asym-metric digraph and association scheme.In Chapter2, we study the relationship between doubly regular asymmetric digraph and non-symmetric association scheme with3or4-classes, and we obtain three relevant theorems.In Chapter3, we first study5different kinds of the doubly regular asymmetric di-graphs, respectively. Then we introduce the construction process of them. Finally, we obtain several kinds of non-symmetric association schemes by using the theorems in the second chapter.