# 臺灣博碩士論文加值系統

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 Let G be a connected multi-graph with vertex set V (G) and edge set E(G).If there exists an edge labeling function f from E(G) to {1,-1}, such that thedifference of numbers of edges labeled 1 and -1 is at most one, then we callsuch f an equitable labeling and G an equitable signed graph.An equitable edge labeling induces a vertex labeling in the following way. Forvertices incident with more 1-edges than (-1)-edges, we label them 1. Forvertices incident with more (-1)-edges than 1-edges, we label them -1. Forvertices incident with the same number of (-1)-edges and 1-edges, we labelthem 0. Then the edge-majority index is de ned as the absolute di erenceof the number of 1-vertices and the number of (-1)-vertices with respect toan equitable edge labeling. The set of all possible edge-majority indices of Gwith respect to all possible equitable labelings is called the edge-majorityindex set of G. Given an equitable edge labeling f of a graph with allodd degrees(all even degrees), we show that all even numbers(all numbers)less than certain edge-majority index with respect to f may be realized bycontinuously switching edge labels.
 1 Introduction 1.1 De nitions 1.2 Background 1.3 Motivation2 Edge-Majority Indices of Odd Graphs 2.1 Basics 2.2 Main Result3 Edge-Majority Indices of Even Graphs 3.1 Basics 3.2 Main Result4 Conclusion and Further Studies
 [1] D. Cartwright and F. Harary, Structural balance: a generalization ofHeider's theory. Psychological Review 63 (1956), 277-293.[2] B.-L. Chen, K.-C. Huang, S.-M. Lee, and S.-S. Liu, On edge-balancedmultigraphs, Journal of Combinatorial Mathematics and CombinatorialComputing, 42(2002), 177-185.[3] Tao-Ming Wang, Chia-Min Lin, and Midge Cozzens, Edge Control inSigned Graphs. Manuscript, 2010.[4] Elliot Kropa, Sin-Min Lee, Christopher Raridan, On the number of unla-beled vertices in edge-friendly labelings of graphs. Discrete Mathematics312 (2012), 574-577.[5] Rosa, A. (1967), On certain valuations of the vertices of a graph, Theoryof Graphs (Internat. Sympos., Rome, 1966), New York: Gordon andBreach, pp. 349-355, MR 0223271.[6] D. Chopra, S-M. Lee and H-H. Su, On edge-balance index sets of wheels,Int. J. of Contemp. Math. Sci., 5 (2010), no. 53, 2605-2620.[7] D. Chopra, S-M. Lee and H-H. Su, On edge-balance index sets of thefans and broken fans, Congr. Numer., 196 (2009), 183-201.[8] M.C. Kong, S-M. Lee and Y.C.Wang, On edge-balance index sets ofsome complete k-partite graphs, Congr. Numer., 196 (2009), 71-94.
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