Circle Circle Of Cartesian Product Figure Edge Pan

With the advent of VLSI circuit technology, it has become possible to builda large parallel and distributed system involving thousands or even tens ofthousands of processors. One crucial step on designing a large-scale parallel anddistributed system is to determine the topology of the interconnection network(network for short).The research of interconnection network is one of the hotspots in researchof parallel computing. A large number of computation problems in network areto use graph embedding problem to simulate and research effectively, such aslooking for efficient data structure in said storage, circuit wiring based on VLSIchip, program structured problems, calculation based on the processor networkorganizations and the problems to determine the bandwidth of the sparse matrixand so on. The embedding of cycles is one of the most important problems ingraph embedding of interconnection network. It can be measured by thepancyclicity and edge-pancyclicity of graph.The Cartesian product graph is an important class of topological structuresof interconnection networks. The Cartesian product graph has recently receivedmuch attention in massively parallel computing systems because it has goodregularity, connectivity, vertex-transitivity, Hamiltonicity etc. The k-ary n-cubenetwork is also a kind of most important networks. On the one hand, manyinterconnection networks can be viewed as the subclasses ofQk include the cycle, the torus and the hypercube. On the other hand, some of massively parallel systems have been built with a k-ary n-cube forming the underlying topology, such as the Cray T3D, J machine and the i Warp.In this paper, we consider the edge-pancyclicity of Cartesian product graphs and the edge-pancyclicity of k-ary n-cubes.In the first chapter, we introduce the graph-theoretical terminology and notation used in our discussion.In the second chapter, we study the edge-pancyclicity of Cartesian product graphs of cycles. We mainly proved that:(2.1) Let k1,k2≥3be two odd integers. Then the torus T(k1,k2)=Ck1×Ck2is (k1+k2/2)-edge-pancyclicity.(2.2) Let k1,k2≥3be two integers and k2is odd. Then the torus T(k1,k2)=Ck1×Ck2is (k1+[k2/2])-edge-pancyclicity(2.3) Let ki≥3be odd for i=1,2,L,n. Then the n-dimensional torus T(k1,k2,L,kn)=Ck1×Ck2×L×Ckn is (?)1/2(ki-1)+1-edge-pancyclicity.(2.4) Let ki≥3be n integers for i=1,2,L,n. Then the n-dimensional torus T(k1,k,L,kn)=Ck1×Ck2×L×Ckn is edge-bipancyclicity.In the third chapter, we study the edge-pancyclicity of Cartesian product graphs of cycles and the edge-pancyclicity of k-ary n-cubes with faulty nodes and edges. We mainly proved that:(3.1) Let m,n be two positive intergers and let G be the two-dimensional torus T2m+1,2n+1with one faulty edge. Then G is (m+n+1)-panconnectivity.(3.2) Let k1,k2,...,kn be n(n≥2) positive intergers and let G be the n-dimensional torus T2k1+1,2k2+1,...,2kn+1with at most2n-3faulty vertices. Then G is (?)(ki+1)-edge-pancyclicity.(3.3) Given an integer n≥2and an odd integer k≥3, let F be a set of k-ary n-cube with fv faulty nodes and fe faulty edges, If fv+fe≤2n-3, then the Qnk-F is (k+1)-edge-pancyclicity.