關於非奇異線圖的樹

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陳資穎(Tzu-Ying Chen) 查詢紙本館藏

畢業系所

數學系

論文名稱

關於非奇異線圖的樹
(On Nonsingular Line Graphs of Trees)

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摘要(中)

本篇論文的目的是在探討刻畫具有非奇異(或奇異)線圖的樹之結構以及檢閱一些用來構造某些具有非奇異(或奇異)線圖的特殊類型之樹的已知結果和基本技術。首先我們證明由k個星形圖Sn1,Sn2,...,和Snk的中心依序連接所形成的樹(Sn1,Sn2,...,Snk)之線圖的奇異性。接著我們研究由k個雙層星形圖S*n1,S*n2,...,和S*nk的中心依序連接所形成的樹(S*n1,S*n2,...,S*nk)之線圖的奇異性。最後我們定義一類特殊的樹Yk(k =1,2,3,...)，這類圖形的定義是遞歸的，首先我們令Y1 = K1,3，然後當我們要建構Yk+1時，就從Yk的每個懸掛點上各加上兩個懸掛邊。在論文的最後一部份我們討論Yk的線圖的奇異性(k =1,2,3,...)。

摘要(英)

The goal of this thesis is to investigate the structures of nonsingular (or singular) line graphs of trees and to review some known results and basic techniques which have been used to obtain the structures of nonsingular (or singular) line graphs of some special classes of trees. First, we show that the singularity of the line graphs of the trees (Sn1, Sn2,..., Snk) obtained by joining the centers of k stars, Sn1,Sn2,..., and Snk , with an edge. And then we show that the singularity of the line graphs of the trees (S*n1, S*n2 ,..., S*nk) obtained by joining the centers of k stars, S*n1, S*n2,..., and S*nk, with an edge. Finally, we define a special class of trees, called Yk graphs (k =1, 2, 3,...). These graphs are defined recursively, let Y1 = K1,3. Next, we construct Yk+1 from the graph Yk by adding two pendent edges to each pendent vertices of Yk. In the last part of the thesis we consider the singularities of L(Yk) for all k.

關鍵字(中)

★ 雙層星形圖
★ 奇異性
★ 線圖
★ 樹

關鍵字(英)

★ Line Graphs of Trees
★ singularity
★ double star

論文目次

Contents i
1 Introduction and preliminaries 1
2 Main results 5
References 17

參考文獻

[1] Bojana Borovicanin and Ivan Gutman, Nullity of Graphs, Mathematics Subject Classi cation (2000): 05C50; 05C90; 92E10.
[2] D. Cvetkovic, I. Gutman, The algebraic multiplicity of the number zero in the spectrum of a bipartite graph, Matematicki Vesnik (Beograd) 9 (1972) 141-150.
[3] R. Grone, R. Merris and V. S. Sunder, The Laplacian Spectrun of a Graph, SIAM Journal of Matrix Theory, 11 (1990) 218-238.
[4] Ivan Gutman, Irene Sciriha, On the nullity of line graphs of trees, Discrete Mathematics 232 (2001) 35-45.
[5] M. C. Marino, I. Sciriha, S. K. Simic, and D. V. Tosic, More about Singular Line Graphs of Trees, Publications de L’’institut Mathematique, 79 (2006) 1-12.
[6] A. Schwenk, Computing the characteristic polynomial of a graph, in: R. A. Bari and F. Harary, ed. Lecture Notes in Mathematics, Graph and Combinatorics, Springer-Verlag 406 (1974) 153-172.
[7] Irene Sciriha, On Singular Line Graphs of Trees, Congressus Numeratium, 135(1998) 73-91.
[8] M. Venkatachalam, N. Mohanapriya and J. Vernold Vivin Star Coloring on Double Star Graph Families, Journal of Modern Mathematics and Statistics 5(1) (2011) 33-36.
[9] Stephen H. Friedberg, Arnold J. Insel and Lawrence E. Spence Linear Algebra (4th Edition), Prentice Hall, 2003.

指導教授

葉鴻國(Hong-Gwa Yeh)

審核日期

2012-7-13

推文