Graph Designs For Graphs Of Nine Vertices With Nine Edges And A 5-Circuit

In this dissertation,we mainly discussed the existence of graph designs,packing designs and covering designs for nine graphs of nine vertices with nine edges and a 5-circuit.For graph design,we first give the necessary conditions,and then the recursive structures are given by using combinatorial design theory.Next,as the basis of recursive structures,a large number of holey graph designs,graph designs and incomplete graph designs need to be constructed.In the process of construction,we used idempotent symmetric Quasi-group,PBD closed sets and difference method,etc.For some graph designs of small orders,we also need to analyze the degree type so as to analyze the possible structures,and further obtain the required designs.Of course,sometimes we need to find the proper algorithms,write programs,and use the high speed of computer operation to find the required constructions.Finally,we proved that the necessary conditions are also sufficient conditions and determined the existence spectrum of graph design.For graph packing design and graph covering design,we proved the existence of optimal packing design and optimal covering design.Meanwhile,we also determined the existence spectrum of a graph with nine vertices,nine edges and a 6-circuit which had not been solved in reference [29],and proved the existence of optimal packing design and optimal covering design.