【作者】 周秀环；

【导师】 李才恒； 潘江敏；

【作者基本信息】 云南大学 ， 基础数学， 2010， 硕士

【摘要】 设Γ为一个图, AutΓ表示Γ的全自同构群。如果AutΓ在Γ的顶点集VΓ和边集EΓ上都是传递的,但在弧集AΓ上不传递,则称图Γ为半传递图。半传递图包含了很多好的性质和例子,吸引了众多学者的关注。例如,徐明耀教授刻画了阶为p3的半传递图[16];D. Maruˇsiˇc刻画了一类正则图和度数为4的半传递图[13];李才恒教授刻画了阶为素数幂的半传递亚循环图[7]。更多的结果参见D. Maruˇsiˇc的综述文章[12]。设G为一个群,一个图Γ称为群G上的Cayley图,如果存在G的一个非空子集S -G{1},使得VΓ= S且顶点x与y邻接当且仅当yx^-1∈S。这个Cayley图记为Cay（G,S）。特别地,亚循环群上的Cayley图称为亚循环图。本文的主要目的是刻画一类半传递亚循环图,并构造一个半传递亚循环图的无穷类。本文所得的主要定理如下:定理1.设G = -a,b- = Zr:Zm是一个中心自由群,这里m = p1p2···ps, s≥2,3 < p1 <···< ps < r为素数,设Γ是G上的X-边传递Cayley图,且度数k < 2p1,其中G- < X≤Aut（Γ）,则下述之一成立:（1）Γ是X-局部本原的。（2）Γ是群G上的正规Cayley图,且X≤Zr:Zr-1是可解的。定理2.假设群G同定理1,k|（r - 1）且k < p1,则G有阶数为k的自同构σ使得bσ= b, aσ= aj。设Γ= Cay（G,S）,则Γ为度数为2k的连通半传递图。更多还原

【Abstract】 LetΓbe a graph, and let AutΓdenote its full automorphism group. If AutΓis transitive on both the vertex-set VΓand edge-set EΓ, but not transitive on thearc-set AΓ, thenΓis called a half-transitive graph. Characterizing and construct-ing half-transitive graphs have received much attention on the literature, see, forexample, half-transitive graphs of prime-cube order is given by Xu[16]; regular half-transitive graphs of valence 4 is constructed by Maruˇsiˇc[13]. Recently, half-transitivemetacirculant graphs of prime-power is characterized by Li[7]. For more results, seeMaruˇsiˇc[12].Let G be a group. A graphΓis called a Cayley graph of G, if there is a subsetS - G, with S = S-1 := {g-1 | g∈S}, such that VΓ= G and x is adjacent to y ifand only if yx-1∈S. This Cayley graph is demoted by Cay(G,S). In particular,Cayley graphs of metacyclic groups are called metacirculants.The main purpose of this thesis is to characterize a class of half-transitivemetacirculants, and to construct an in-nite family of connected half-transitive metacir-culants. The main results are the following two theorems.Theorem 1. Let G = -a,b- = Zr:Zm be a center-free group, where m = p1p2···pswith s≥2 and 3 < p1 <···< ps < r are primes. LetΓbe an X-edge-transitiveCayley graph of G of valency k < 2p1, where G- < X≤AutΓ. Then either(1)Γis X-locally primitive, or(2)Γis a normal Cayley graph of G and X≤Zr:Zr-1 is soluble.Theorem 2. Let G be a group as in Theorem 1. Suppose that k|(r - 1), k < p1.Then there existsσ∈Aut(G) such that o(σ) = k, bσ= b and aσ= aj. Let Γ= Cay(G,S), where ThenΓis a connected half-transitive graph of valency 2k.更多还原