Research On Group Connectivity Of Some Special Graphs

In1950, Tutte leaded to the concept of nowhere-zero flows in order to studygraph coloring problem, and made a lot of conjecture, get some basic conclusions. In1992, Such questions were summarized by Jaeger et al. who further proposed theconcept of connectivity group. The study of nowhere-zero flow and groupconnectivity helped to solve the four-color conjecture coloring problem; In addition,the study of this issue help us to deeper understand the network structure in realnetworks. This paper study two issues: the first one is nowhere-zero flow of regulargraphs; the second on is crystal graph and cartesian product graph of circle and roadgroup connectivity. This paper worked the following work focusing on these twoissues.Firstly, we ideas to the point of the graph through application programming andoperator the assignment by the defined of group connectivity, relying on the loopalgorithm to get a conditions that odd-order regular graph has aZ2m NZF,furthermore we proofed that the graph has a odd circle that circle degree point is threeits has noZ3nowhere-zero flow.Then, we proved the group connectivity of crystal graph of by deleting andshrinking two edges adjacent points that the degree is three.Finally, we obtained sub-graph by dividing the graph, the first part of the whole,using the proof by contradiction, and by the reasonably assignment of points, weproved that cartesian product graph of odd-order circle and road group connectivity is4and cartesian product graph of even-order circle and road group connectivity lessthan4.In sum, the graph can contain all the structural features of networks, so researchthe structural features of networks was to research the features of graphs. Regulargraph, crystal graph, cartesian product graph were special graphics in graph theory, so,clearly researching nowhere-zero flows and group connectivity problem can pave theway for the study of general graphics. In view of this, some basic work that thenowhere-zero flows of regular graphs and cartesian product graph of circle and roadgroup connectivity were done.