令G為含有m個邊的簡單圖，若存在一個函數f，其中f : V(G)→{0,1,2,…,m}且f為一對一函數。若由f 所衍生出的函數 g，g : E(G)→{1,2,…,m}，∀e={u,v}∈E(G)，g(e)=|f(u)-f(v)| 且g 為一對一且映成函數，則稱此圖G為優美圖。
在本論文中，我們證明了： 
(1)利用PnvKm是優美圖，獲得設m、n為正整數，若G為n個點，n-1個邊的優美圖，則GvKm為優美圖。
(2)若t為正整數，則當m=3,4,5時，CmvKt為優美圖。

Let G be a simple graph with m edges. If there is a function f : V(G)→{0,1,2,…,m} and fis one-to-one. From the function f we can get a function 
g : E(G)→{1,2,…,m}, defined by g(e)=|f(u)-f(v)|, for e={u,v}∈E(G), and g is bijective, then we call f is a graceful labeling and G is a grace graph.
In this thesis, we obtain the following results.
(1)Let m and n be positive integers. If G is a graph with n vertices and n–1 edges and G is graceful, then GvKm  is a graceful graph.
(2)Let t be appositive integer. If m=3,4,or 5,then CmvKtis a graceful graph.

目錄
目錄...............................i
圖目錄............................ii
第一章  簡介.........................1
第二章  預備知識.....................3
第三章  主要內容....................10
    第一節  點與點相連的圖..........10
    第二節  路徑與點相連的圖........12
    第三節  迴圈與點相連的圖........16
第四章  結論與發展..................28
參考文獻............................29
圖目錄
圖2.1  圖G3.........................3
圖2.2  deg(v)=3.....................4 
圖2.3  圖G..........................4
圖2.4  路徑P5......................4
圖2.5  連通圖G1與非連通圖G2........5
圖2.6  迴圈........................5  
圖2.7  完全圖......................6
圖2.8  圖G與圖G的補圖..............6
圖2.9  完全二分圖Km,n.............7
圖2.10 P3vP2......................7
圖2.11 輪圖W7.....................8
圖2.12 輪圖W7的點標號.............8
圖2.13 K3的優美標號...............9
圖3.1 Km,n.......................11 
圖3.2 K4,4與K4,5的優美圖.........11
圖3.3 Pn.........................12
圖3.4 P5,P6的優美標號............13
圖3.5 P3vK1,P4vK2的優美標號......13
圖3.6 PnvKm......................14
圖3.7 C3vK1及C3vK1的優美標號.....16
圖3.8 C3vK2及C3vK2的優美標號.....17
圖3.9 C3vK3及C3vK3的優美標號.....17
圖3.10 C3vKt.....................18
圖3.11 C3vK9的優美標號...........19
圖3.12 C4vK1及C4vK1的優美標號....20
圖3.13 C4vk2及C4vK2的優美標號....20
圖3.14 C4vKt.....................21
圖3.15 C4vK8的優美標號...........23
圖3.16 C5vK1及C5vK1的優美標號....23
圖3.17 C5vK2及C5vK2的優美標號....24
圖3.18 C5vK3及C5vK3的優美標號....24
圖3.19 C5vKt.....................25
圖3.20 C5vK7的優美標號...........27

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