### Title page for etd-0127111-064740

URN etd-0127111-064740 Pei-Lan Yen No Public. This thesis had been viewed 5202 times. Download 672 times. Applied Mathematics 2010 1 Ph.D. English A study of convexity in directed graphs 2011-01-19 70 convexity spectrum ratio difference of convexity spectra strong convexity spectrum convexity spectrum convex set orientable convexity number convexity number Convexity in graphs has been widely discussed in graph theory and G.Chartrand et al. introduced the convexity number of oriented graphsin 2002. In this thesis, we follow the notions addressed by them anddevelop an extension in some related topics of convexity in directedgraphs. Let D be a connected oriented graph. A set S subseteq V(D)is convex in D if, for every pair of vertices x, yin S,the vertex set of every x-y geodesic (x-y shortest directedpath) and y-x geodesic in D is contained in S. The convexity number con(D) of a nontrivial oriented graph D isthe maximum cardinality of a proper convex set of D. We show thatfor every possible triple n, m, k of integers except for k=4,there exists a strongly connected digraph D of order n, sizem, and con(D)=k. Let G be a graph. We definethe convexity spectrum S_{C}(G)={con(D): D is anorientation of G} and the strong convexity spectrumS_{SC}(G)={con(D): D is a strongly connected orientation ofG}. Then S_{SC}(G) ⊆ S_{C}(G). We show that for anyn ≠ 4, 1 ≤ a ≤ n-2 and a n ≠ 2, there exists a2-connected graph G with n vertices such thatS_C(G)=S_{SC}(G)={a,n-1}, and there is no connected graph G oforder n ≥ 3 with S_{SC}(G)={n-1}. We also characterizes theconvexity spectrum and the strong convexity spectrum of completegraphs, complete bipartite graphs, and wheel graphs. Those convexityspectra are of different kinds.Let the difference of convexity spectra D_{CS}(G)=S_{C}(G)-S_{SC}(G) and the difference number of convexity spectra dcs(G) be the cardinality of D_{CS}(G) for a graph G. We showthat 0 ≤ dcs(G) ≤ ⌊|V(G)|/2⌋,dcs(K_{r,s})=⌊(r+s)/2⌋-2 for 4 ≤ r ≤ s,and dcs(W_{1,n-1})= 0 for n ≥ 5.The convexity spectrum ratio of a sequence of simple graphsG_n of order n is r_C(G_n)= liminflimits_{n to infty}frac{|S_{C}(G_n)|}{n-1}. We show that for every even integer t,there exists a sequence of graphs G_n such that r_C(G_n)=1/t anda formula for r_C(G) in subdivisions of G. Ko-Wei Lih - chair Dah-Jyh Guan - co-chair Gerard Jennhwa Chang - co-chair Xuding Zhu - co-chair Sen-Peng Eu - co-chair Tsai-Lien Wong - co-chair Hong-Gwa Yeh - co-chair Li-Da Tong - advisor indicate access worldwide 2011-01-27

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