The CI-Property Of Coset Graphs

Group theory is a science of researching symmetry property, among which group action especially the action of group on symmetry combined structure has been being a very hot issue. The action of group on graphs is the most fundamental and natural one, so this association of group and graph starts wider field for group theory research. On studying the group and graph, some graphs being a very good symmetry property can be obtained by group. The classical representations of these type graphs are Cayley graphs and Sabidussi Coset graphs.People research more about the Cayley graph of group, among which the isomorphism problem of Cayley graph, i. e. the CI-property of Cayley graph, is originated from a Adam's conjecture in 1967. In the last four decades, there are a lot of scholars exploring deeply this question and obtaining abundant results. The representative persons are Adam, B. D. Mckay, M. Muzychuk, G. Royle, M. Conder, Praeger, M. Y. Xu, C. H. Li and so on.Coset graph is more common than Cayley graph. Owing to every vertex transitive graphs may not be a Cayley graphs but be a Coset graphs, the research about the symmetry property of Coset graphs is more significant than the Cayley graphs. Similar to the CI-property of the Cayley graph, we can also define the CI-property of the Coset graph. Having not seen the researching results of this part is a motion of researching the CI-property of Coset graph in this paper. This paper chooses some special groups to research the CI-property of the Coset graph and obtain some results. This paper is arranged as follows:In Chapter one, we describe the definition of the CI-property of the Coset graph and some basic results;In Chapter two, we research the CI-property of the Coset graph in qp order group;In Chapter three, we research the Cl-property of the Coset graph in some groups, for instance, alternative group A₄, the least order simple group A₅, dihedral group D₈, Hamilton .group Q₈ and a special group.