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系統識別號 U0026-0107201523385500
論文名稱(中文) 有限環的零因子圖
論文名稱(英文) On The Zero-divisor Graphs of Finite Rings
校院名稱 成功大學
系所名稱(中) 數學系應用數學碩博士班
系所名稱(英) Department of Mathematics
學年度 103
學期 2
出版年 104
研究生(中文) 郭子揚
研究生(英文) Tzu-Yang Kuo
學號 L16021080
學位類別 碩士
語文別 英文
論文頁數 22頁
口試委員 指導教授-蕭仁傑
口試委員-黃柏嶧
口試委員-黃世昌
中文關鍵字 零因子圖  完全圖  矩陣  鄰域 
英文關鍵字 zero-divisor graph  complete  matrix ring  closed neighbourhood 
學科別分類
中文摘要 這篇碩士論文裡,我們首先討論有限環的零因子圖。結果顯示一個零因子圖的導出子圖是完全圖且比原來的零因子圖少一個點的話,那麼原來的零因子圖就必須是完全圖。另外確認了零因子圖一定不包含某種特殊的導出子圖。章節最後這些結果被用來確認所有包含五個點的有限交換環的零因子圖種類的可能性。
第二個部分我們研究關於有限交換環上矩陣的零因子圖。矩陣本身是非交換環,其零因子圖是有向圖。最後,對於一個點的鄰域,我們對其繪製零因子圖並得到一些有趣的結果。其圖必定連通而且直徑不大於四。
英文摘要 In this thesis, we first consider the zero-divisor graph of a finite commutative ring $R$. We show that if there is a complete subgraph of the zero-divisor graph Γ(R), which is obtained by deleting a vertex from Γ(R), then Γ(R) is complete. We also find that Γ(R) admits no subgraphs of certain special type. These results are used to determine all possible zero-divisor graphs of finite commutative rings with five vertices.
The second part of this thesis studies the zero-divisor graphs Γ(Mn(R)) of matrix rings over finite commutative ring, Mn(R), which is a directed graph. In particular, we found some interesting results about the closed neighbourhood of a given vertex in Γ(Mn(R)).
論文目次 1. Introduction ---------------------------------------------------- 1
2. Subgraphs of special types ----------------------------- 3
3. Graphs with five vertices --------------------------------- 6
4. Zero divisor graphs of matrix rings ------------------ 10
4.1. The subgraph of the closed neighbourhood --- 11
4.2. The cardinality of N[A] ---------------------------- --- 13
5. Appendix ------------------------------------------------------ 19
References ------------------------------------------------------- 22
參考文獻 [1] D. D. Anderson and M. Naseer. Beck’s coloring of a commutative ring. J. Algebra, 159(2):500–514, 1993.
[2] David F. Anderson, Andrea Frazier, Aaron Lauve, and Philip S. Livingston. The zero-divisor graph of a commutative ring. II. In Ideal theoretic methods in commutative algebra(Columbia, MO, 1999), volume 220 of Lecture Notes in Pure and Appl. Math., pages 61–72.Dekker, New York, 2001.
[3] David F. Anderson and Philip S. Livingston. The zero-divisor graph of a commutative ring. J. Algebra, 217(2):434–447, 1999.
[4] Istv´an Beck. Coloring of commutative rings. J. Algebra, 116(1):208–226, 1988.
[5] Ivana Boˇzi´c and Zoran Petrovi´c. Zero-divisor graphs of matrices over commutative rings. Comm. Algebra, 37(4):1186–1192, 2009.
[6] William C. Brown. Matrices over commutative rings, volume 169 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, Inc., New York, 1993.
[7] Frank Harary and Edgar M. Palmer. Graphical enumeration. Academic Press, New York-London, 1973.
[8] Shane Patrick Redmond. Generalizations of the zero-divisor graph of a ring. ProQuest LLC, Ann Arbor, MI, 2001. Thesis (Ph.D.)–The University of Tennessee.
[9] Haruo Yanai, Kei Takeuchi, and Yoshio Takane. Projection matrices, generalized inverse matrices, and singular value decomposition. Statistics for Social and Behavioral Sciences. Springer, New York, 2011
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