Research On The Problem Of Fractional Matching Preclusion Of Data Center Network And Some Cartesian Product Graphs

It plays a crucial role for the fault tolerance of a network that the network diagnos-tic system effectively identifies,repairs,and replaces faulty processors in the system.An ideal network system should be secure and reliable.Furthermore,the topology structure of a network is usually modeled as a connected graph,so that the fault tol-erant questions of a network can be transformed into the related questions of graph theory.Among them,the matching exclusion of graph theory is a question that can measure the fault tolerance of networks.The concept of matching preclusion was formally introduced and defined by Brigham et al.in 2005.Let F be the edge set of graph G.If G-F has no per-fect matching or almost perfect matching,then the number of minimum edge set F is called the number of conditional matching preclusion of graph G,denoted as mp(G).With the further deepening of research,Cheng,et al.defined the concept of condi-tional matching preclusion.Let F be the edge set of graph G.If G-F has neither isolated vertices nor perfect matching or almost perfect matching,then the number of minimum edge set F is called the number of conditional matching preclusion of graph G,denoted as mp₁(G).Subsequently,Liu Yan et al.in 2017 proposed the concepts of fractional matching preclusion number and fractional strong matching preclusion number.Let F be an edge subset of a graph G.If G-F has no fractional perfec-t matching,then the minimum number of edges of F is called fractional matching preclusion number,denoted as f mp(G).Let F be a subset of edges and vertices of a graph G.If G-F has no fractional perfect matching,then the minimum number of F is called fractional strong matching preclusion number,denoted as f smp(G).The data center network is a very important interconnection network.It can sup-port millions of servers with high network capacity only by using commercial switch-es.In this thesis,we firstly study the FMP number and FSMP number of data cen-ter networks.The fractional matching preclusion sets and fractional strong matching preclusion sets are classified.The Cartesian product is an important structure of the interconnection network.Next,we study the fractional strong matching preclusion of Cartesian product of the graph and the cycle,give FSMP number for this type of graph and prove that they are super fractional matching.Then,we study FMP num-ber of torus networks and prove that it is super fractional matching;and use the result of Cartesian product of the graph and the cycle to deduce the FSMP number of torus networks and fractional strong matching preclusion sets are classified.The exchange crossed cube has many good topological properties.In the end,we study MP num-ber and conditional matching preclusion number and properties of super(conditional)matching of exchange crossed cube.