Some Properties Of Cayley Graph On Inner 2-Closed Groups With Order 4p

Cayley graph was defined by A. Cayley in 1878 only for explaining the generating elements and their definition relations. With its simple construction, high symmetry and so much variety, the Cayley graph has been paid more and more attention in studying groups and graphs. It is also the main purpose of my thesis.Isomorphism problems on the Cayley graph are fundamental in its researching fields and also very complicating. First, we need to investigate if some different Cayley graphs of the same group are isomorphic each other. Secondly, some Cayley graphs of the differentgroups may be isomorphic (an extreme example is a complete graph of n vertices which is a Cayley graph on any group with order n) [1].Up to date,the researching is concentrated on the former problem and mainly on the so-called CI-property. As a matter of fact, the DCI-groups are rather rare [2, 3, 4, 5], so that the researching was turned to the m-DCI-property and m-CI-property [6, 7, 8], even weak m-DCI-property and weak m-CI-property [8, 9, 10]. This thesis mainly study m-DCI-property of group G = (a, b | a^2α = b^p = 1, a^-1ba = b^-1)(m = 1,2,3;α = 2) and weak m-CI-property(m = 4,5; α = 2;p = 3).Particularly,we prove that inner 2-closed group with order 12 is weak CI-group. In fact, when α= 1, |G| = 2p, Babai proofed that groups with order 2p are DCI-group in 1977. While when α = 2, group G with order 4p that is generalized double cyclic group. We proved that G is 3-DCI-group and weak 5-CI-group.The study in symmetry of graphs and classfication on vertex-transitive graph have always been hot in recent years [11, 12]. Cayley graphs act as an important kind of vertex-transitive graph. But in those problems study, we must often determine full automorphism of corresponding Cayley graph [13]. The regularity problem of Cayley graph is also a problem that shoud be more deeply studied after the solution of the problem of a group's DRR [14]. The thesis here really give a complete classficationof the normal connected undirected graphs with 4 valencies on inner 2-closed group with order Ap. In fact,they are all infinite families of normal connected undirected graphs with 4 valencies.For the case of 6 valencies, we also give a complete classfication of the normal connected undirected graphs on inner 2-closed groups with order 12. Moreover for the case of group with 4p, we also give some infinite families of normal connected...