本文選題：對合交換圖 + k-正則圖　； 參考：《湖北民族學(xué)院》2017年碩士論文

【摘要】：對合交換圖是以群的一個二階元共軛類為頂點,兩頂點有邊當(dāng)且僅當(dāng)它們交換的圖.本文首先研究了對合交換圖為0-正則和1-正則圖時群結(jié)構(gòu)的相關(guān)問題.其次,本文討論了一些特殊類群對合交換圖的結(jié)構(gòu).全文由四章組成.第1章介紹了對合交換圖的研究背景、相關(guān)的定義、全文涉及的基本理論及基本符號.第2章討論了0-正則和1-正則對合交換圖一些相關(guān)的結(jié)構(gòu)、性質(zhì),并給出了對合交換圖為0-正則圖的一個充要條件.主要證明了若有限群G的對合交換圖)(IG(38)為0-正則圖,并且G(28)?I?,則G有一個指數(shù)為2的奇階子群.最后,我們給出了對合交換圖為1-正則圖的若干群例及兩個猜想.第3章研討了亞循環(huán)2-群對合交換圖的結(jié)構(gòu),主要證明了亞循環(huán)2-群的對合交換圖為0-正則或1-正則圖.第4章討論了亞交換群對合交換圖的結(jié)構(gòu),得出對于交換群A和B,若G(28)AfBExt),;,(?,?),(a為G中任意二階元,|{??)((10)-(28)bbH Ab?}為A的子群,則G關(guān)于?),(a的對合交換圖是r)1(--正則圖,其中r為所有陪集?)(),(),(??????-(10)-(10)aaffH,??B?中二階元的個數(shù).最后,我們給出了一個具體計算亞交換群對合交換圖的例子,即28?(28)ZZA,22?(28)ZZB,且G(28)AfBExt),;,(?,計算出所有的對合交換圖的度,并得出G的對合交換圖為0-正則、1-正則或2-正則圖.
[Abstract]:Involutive commutative graph is a class of second order conjugate of group as vertex, two vertices have edge if and only if they commutate. In this paper, we first study the problem of group structure when involutive commutative graphs are 0-regular and 1-regular. Secondly, we discuss the structure of some special class groups involutive commutative graphs. The full text consists of four chapters. Chapter 1 introduces the background, definition, basic theory and symbol of involutive commutative graph. In chapter 2, we discuss some related structures and properties of 0-regular and 1-regular involutive commutative graphs, and give a necessary and sufficient condition for involutive commutative graphs to be 0-regular graphs. It is proved that G has an odd order subgroup if the involutive commutative graph of finite group G is a 0-regular graph and G ~ (2) is an odd order subgroup. Finally, we give some group examples and two conjectures that involutive commutative graphs are 1-regular graphs. In chapter 3, we discuss the structure of subcyclic 2-group involutive commutative graphs, and prove that the involutive commutative graphs of subcyclic 2-groups are 0-regular or 1-regular. In chapter 4, we discuss the structure of the involutive commutative graphs of subabelian groups A and B, and obtain that for the abelian groups A and B, if G is an arbitrary second order element in G, and G is a subgroup of A, then G's involutive commutative graph about A is a rtl ~ (1) -regular graph, where r is all the guest set ~ (-10) -10 ~ (aaffHB) ~ (?) _ B _ (?) _ _ _ The number of middle and second order elements. Finally, we give an example of calculating the involutive commutative graph of subabelian group, that is, 28AZA22 / 28ZZB, and GX28 / AfBExtl, calculate the degree of all involutive commutative graphs, and obtain that the involutive commutative graph of G is 0-regular 1-regular or 2-regular graph, and that the involutive commutative graph of G is 0-regular 1-regular or 2-regular graph, and the involutive commutative graph of G is 0-regular 1-regular or 2-regular.
【學(xué)位授予單位】：湖北民族學(xué)院
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O152.1

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本文編號：2005450