The Super Edge-connectivity And Restricted Edge-connectivity Of Hypergraphs

The edge-connectivity of graphs is an important parameter to measure networks.When the edge-connectivity is used as a measure,the most reliable networks correspond to two classes of graphs:maximally edge-connected graphs and super edge-connected graphs.In order to quantitatively characterize the edge-connectivity of graphs,Esfahanian and Hakimi proposed the restricted edge-connectivity of graphs in 1988.The hypergraph is a natural extension of the graph.It can be used to study the problem of multiple subsets,analyze the multiple relationships among the elements in a finite set,and describe the most general discrete structure relationship.In the field of computer science,if the processor is regarded as a vertex,and the bus is regarded as an edge of a hypergraph,a multi-processor system can be regarded as a hypergraph.Researchers have given many results on the edge-connectivity and restricted edge-connectivity of graphs.However,the corresponding results on hypergraphs is still relatively few.This thesis is divided into three chapters to study the edge-connectivity and restricted edge-connectivity of hypergraphs.Chapter 1,we introduce some basic concepts and symbols of graph theory that will be used,and give the research background,main concepts and main results of the thesis.Chapter 2,we first describe the relationship between the super edge-connectivity and maximally edge-connectivity of hypergraphs,and prove that intersecting hypergraphs must be super edge-connected.Then,the concept of complete graphs is extended to hypergraphs,and some propertyies of them are given.Moreover,minimum degree conditions and diameter conditions of super edge-connected hypergraphs are also given,and the optimality of these conditions is illustrated by examples.Chapter 3,the concepts of the restricted edge-connectivity and maximally restricted edge-connectivity of hypergraphs are generalized to hypergraphs.First,a sufficient con-dition for restricted edge-connected hypergraphs and an upper bound on the restricted edge-connectivity are given.Then the optimality of the results are illustrated by examples.Furthermore,minimum degree conditions and diameter conditions of maximally restricted edge-connected hypergraphs are given.