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Some Properties of Cayley Graph on Inner 2Closed Groups with Order 4p
Author: ZhangCui
Tutor: XuShangJin
School: Guangxi University
Course: Basic mathematics
Keywords: inner 2closed group Cayley graph Graph isomorphism DCIproperty normalproperty DRR Hamiltonproperty egdeHamiltonproperty
CLC: O157.5
Type: Master's thesis
Year: 2006
Downloads: 23
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Abstract
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 socalled CIproperty. As a matter of fact, the DCIgroups are rather rare [2, 3, 4, 5], so that the researching was turned to the mDCIproperty and mCIproperty [6, 7, 8], even weak mDCIproperty and weak mCIproperty [8, 9, 10]. This thesis mainly study mDCIproperty of group G = (a, b  a^{2α} = b^{p} = 1, a^{1}ba = b^{1})（m = 1,2,3;α = 2） and weak mCIproperty（m = 4,5; α = 2;p = 3）.Particularly,we prove that inner 2closed group with order 12 is weak CIgroup. In fact, when α= 1, G = 2p, Babai proofed that groups with order 2p are DCIgroup in 1977. While when α = 2, group G with order 4p that is generalized double cyclic group. We proved that G is 3DCIgroup and weak 5CIgroup.The study in symmetry of graphs and classfication on vertextransitive graph have always been hot in recent years [11, 12]. Cayley graphs act as an important kind of vertextransitive 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 2closed 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 2closed groups with order 12. Moreover for the case of group with 4p, we also give some infinite families of normal connected

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CLC: > Mathematical sciences and chemical > Mathematics > Algebra,number theory, portfolio theory > Combinatorics ( combinatorics ) > Graph Theory
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