給定任意圖G=(V,E)，|V(G)|=n， |E(G)|=m，圖G的一個標號f:E→{1,2,…,m}為一雙射函數。假設對於任意頂點u，令
s(u)=∑_(e∈E(u))〖f(e)〗，其中E(u)為所有與u相連的邊的集合，若對於任意i≠j，s(i)≠s(j)，則稱此標號為反魔方標號。假設f是一個圖G的反魔方標號，令deg⁡(u)為所有與u相連的邊的數量總和，若對於任意兩個不同的頂點u,v，deg⁡(u)<deg⁡(v)，滿足s(u)<s(v)，則稱f為圖G的一個強反魔方標號。

在本文中我們想要討論的是雙蜘蛛圖。由於此圖的部分形式被證明有反魔方，因此我們要證明對於任意的雙蜘蛛圖都有強反魔方標號。

Let G=(V,E) be a simple graph with n vertices and m edges. A labeling of G is a bijection from the set of edges to the set {1,2,…,m} of integers, for each vertex, its vertex sum is defined to be the sum of labels of all edges incident to it. If all vertices have distinct vertex sums, we call this labeling anti-magic. Suppose f is an anti-magic labeling of G, and for any two vertices u,v with deg(u) < deg(v), if vertex sum of u is strictly less than vertex sum of v, then we say f is a strongly anti-magic labeling of G. 

In this thesis, we restrict our graphs to double spider graphs. Since some of double spider graphs have already been proven to be anti-magic, we will prove a stronger result here, that is, all double spider graphs are strongly anti-magic.

目錄 …………………………………………………………I
圖表目錄 ……………………………………………………II
第一章 基本定義……………………………………………1
第二章 反魔方標號…………………………………………5
第三章 雙蜘蛛圖的強反魔方標號…………………………9
參考文獻 ……………………………………………………35

圖表目錄
圖1.1(路徑) …………………………………………………1
圖1.2(迴圈) …………………………………………………2
圖1.3(樹圖) …………………………………………………2
圖1.4(蜘蛛圖) ………………………………………………3
圖1.5(雙蜘蛛圖) ……………………………………………4
圖2.1(P_7) ……………………………………………………5
圖2.2(S(4,4,3,3,2,2,2))………………………………………7
圖3.1…………………………………………………………14
圖3.2…………………………………………………………16
圖3.3…………………………………………………………18
圖3.4…………………………………………………………21
圖3.5…………………………………………………………24
圖3.6…………………………………………………………27
圖3.7…………………………………………………………29
圖3.8…………………………………………………………30
圖3.9…………………………………………………………30
圖3.10 ………………………………………………………31
圖3.11 ………………………………………………………32
圖3.12 ………………………………………………………32
圖3.13 ………………………………………………………33

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