旧貌换新颜 —— 从 河图 到 魔图 的创新之路
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旧貌换新颜 —— 从 河图 到 魔图 的创新之路. 李信明（笔名李学数） 美国圣荷西州立大学教授. Lee Sin-Min (Xue Shu). 数学工作者及科普作家，出版一系列的 《 数学和数学家的故事 》 及 《 中国数学五千年 》 。 他是前“香港数学教师协会”的顾问，也是一个“数学传道士”，喜欢与不同年龄的人谈数学。 这个演讲把听众从中国古代的 河图 的历史演变带领到今日“图论”，一个活跃的研究领域。 讲者介绍他开创图论里的“标号理论”的一个分支的故事。听众不需具备数学的专业知识，只要有中学的知识水平就能了解内容。 - PowerPoint PPT Presentation
Transcript of 旧貌换新颜 —— 从 河图 到 魔图 的创新之路
旧貌换新颜——从河图到魔图的创新之路
李信明（笔名李学数）美国圣荷西州立大学教授
Lee Sin-Min (Xue Shu)
• 数学工作者及科普作家，出版一系列的《数学和数学家的故事》及《中国数学五千年》。
• 他是前“香港数学教师协会”的顾问，也是一个“数学传道士”，喜欢与不同年龄的人谈数学。
• 这个演讲把听众从中国古代的河图的历史演变带领到今日“图论”，一个活跃的研究领域。
• 讲者介绍他开创图论里的“标号理论”的一个分支的故事。听众不需具备数学的专业知识，只要有中学的知识水平就能了解内容。
• 对于想从事数学研究的人，这演讲会提供许多值得思考的新课题。
• 一个数学家必须是在每个星期都有一些新的研究工作才成为数学家。 —— Paul Erdos (1913-1996)
1984 年，他获得 Wolf 奖
「山海经」中说「伏羲得河图，夏人因之，曰《连山》」。
《易 . 系辞上》：“是故天生神物，圣人则之；天地变化，圣人故之；天垂象见吉凶，圣人象之；河出《图》，洛出《书》，圣人则之。”
• 北方：一个白点在内，六个黑点在外，表示玄武星象，五行为水。 东方：三个白点在内，八个黑点在外，表示青龙星象，五行为木。 南方：二个黑点在内，七个白点在外，表示朱雀星象，五行为火。 西方：四个黑点在内，九个白点在外，表示白虎星象，五行为金。 中央：五个白点在内，十个黑点在外，表示时空奇点，五行为土。
距今七、八千年之伏羲时代，一龙马从黄河跃出，其身刻有“一六居下，二七居上，三八居左，四九居右”之数字。此为河图。
伏羲依照河图而演绎为八卦
易经系辞传曰：「河出图，洛出书，圣人则之。」
• 自伏羲发现河图，这以后差不多过了八百年，当时洪水泛滥，百姓流离失所，大禹临危受命，婚后第四天就率众治水，并且三过家门而不入。各种方法都用过了，大禹始终没有找到治水的良策，后来有一天，他发现一只五色彩龟出现在洛水，背上的纹理形态如同文字，就此，大禹就发现了「洛书」了。 如今在洛阳孟津，修了一座巍峨的龙马负图寺。寺殿距诸侯会盟处不远。
• 目前我国学术界公认的最早记载了河图、洛书的书籍是西周初期王宫史官记载而成的《尚书》，也就是后世所称的《书》经。其中《尚书》《顾命》篇中记载到：周武王的儿子周成王执政十九年后病逝，西周史官在周成王之子周康王于洛邑文王太庙大室中举行的继位典礼上看到：“越玉、五重、陈宝、赤刀、大训、弘璧、琬琰在西序；大玉、夷玉、天球、河图、洛书在东序”。
纵横图• 古代人称为“纵横图”，是指在一个 n×n
的方阵上排上 1， 2， 3，…… n2的数，使得每一行、每一列以及主对角线上的数的和是一样。这个叫法是在十三世纪南宋的数学家杨辉的称呼，西洋人都成为魔方阵，以为他们是具有法力的。
洛書上的 3 階縱橫圖戴九履一、左三右七、二四為肩、六八為足、五位居中
南宋数学家杨辉 (1433-1483) ，在他著的《续古摘奇算法》里介绍了这种方法：只要将九个自然数按照从小到大的递增次序斜排，然后把上、下两数对调，左、右两数也对调；最后再把中部四数各向外面挺出，幻方就出现了。 创“纵横图”之名
• 在我国现存的最早的天文历算著作《周髀算经》中有这样一句话：“洛书者，圆之象也”。洛书使用数字构造出的一个“圆之象”。宋人的解释是：洛书横、竖、斜的数之和都是十五，九个数的和是四十五，是十五的三倍，符合“圆者一围三”。还有一个解释是：洛书不管怎样摆，不管从那个角度数，其直线上的和都是一个固定数，而这正是圆的直径的特点。
洛書上的 3 階縱橫圖1977 年春 , 安徽省阜陽縣城郊的衣民在雙古堆平整土地時 ,
發現了兩座古墓 . 文物工作者發掘後證明這是西漢汝陽侯的墓葬 . 夏侯灶死於漢文帝 15 年 , 即公元前 165 年 , 距今已 2170 多年 . 出土文物中包含 3 件極為珍貴的中國古代天文儀器 , 其中一件叫”太乙九宮占盤” , 是用來占卦的盤 , 分上盤和下盤兩部分 , 上盤嵌入下盤的凹槽 , 可以隨意轉動 , 如圖所示 . 將盤上的古漢字轉寫成現代漢字以後如圖 . 由圖可見 , 太乙九宮占盤正面是按八卦位置和金 , 木 ,水 , 火 , 土五行屬排列的 , 其九宮名稱和各宮節氣的天數與古書”樞經”完全一致 . 這個占盤就是用來測算立春 , 春分 , 立夏 , 夏至 , 立秋 , 秋分 , 立冬 , 冬至這八個節氣的 .我們感興趣的是盤上圓圈中 8 個方位上的數字如果 ? 上中心因安裝轉軸而無法刻上的” 5” 的話 , 恰為九宮數字”四九二 , 三五七 , 八一六” !
楊輝除以”楊輝三角”聞名於世外 , 也是世界上第一個從數學角度對幻方進行詳盡研究的學者 , 並取得了豐碩成果 . 他在 1275 年所著的”續古摘奇算法”兩卷中 , 除了給出洛書中 3 階幻方的構造方法以外 , 還給出了4 階至 10 階的幻方 , 其中 4 階至 8 階幻方各給出兩圖 , 楊輝稱之為陰 , 陽圖 .
钱塘 ( 今杭州 ) 人 , 生平履历不详 , 生活于 13世纪。 1261 年所著的《详解九章算法》一书中，辑录了三角形数表，称之为“开方作法本源”图
對洛書上的 3 階幻方 , 楊輝將其生成法和最後佈局歸結為以下 8 句話
九子斜排 上下對易 左右相更 四維挺出戴九履一 左三右七 二四為肩 六八為足
階梯法也叫座樓梯法 , 是法國數學家巴赫特創造的 . 這個方法把 n 階方陣四周向外擴展成階梯狀 , 然後把 1~n2 個自然數順階梯方向先碼放好 , 再把方陣以外部分平移到方陣以內其對邊部分中去 , 即構成幻方 . 這個方法十分簡單而巧妙 ,適用於所有奇數階幻方 .
首先將奇數沿菱形的各平行邊順序排列好 , 然後將偶數沿這些邊的延長線的方向順序填入空格 ,回到這條邊的起點時 , 轉移到下一條邊 , 如此循環往返 , 就可構成幻方 . 圖中 ,奇數 1,3,5 這邊上填入 2,4 而 7,9 這邊上則填入了 6,8,10, 如此等等 .
楊輝的這個幻圓是將自然數 1~33 分布在 4 個同心圓與 4條直徑的交點上 , 而將 9 置於中心 ,? 圖的名稱由此而來 . 在圖中 ,4條直徑上 9 個數的和都等於 147,8條半徑上 4 數支和都等於 69.
由吳文俊院士主編的”中國數學史大系”對楊輝的圖給予了極高評價 ,認為他是“史無前例”的 .實際上圖還有一個奇異性質 : 他的垂直直徑的上半截與他右側 3條半徑上的16 個數正好組成一個半幻方 ; 而垂直直徑下半截與他左側3條半徑上的 16 個數也正好組成一個半幻方 ,及每行每列4 數之和都是 69,但對角線一為 70, 一為 68.
聚六圖
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聚 6 圖在正六邊形的 6 個頂點處各有 6 個小圓圍成一圈 , 內中 6 數之和均為 111, 楊輝謂之”六子回環 , 各一百一十一” .但新加坡學者藍麗蓉認為 , 這個聚 6 圖在轉抄 ,復刻過程中發生了錯誤 , 與六邊形左右頂角相接的小圓不應是 31 與 24, 而應該是 8 與 1, 與右上頂角相接的也不應是 7 而應是 28,此外 , 右下角左側 3個小圓位置也有錯誤 , 從上到下應分別為 4,19,23,這樣改正以後的聚 6 圖 , 就不單有 6 組 6 數之和為幻和 111. 內層和外層 6 數之和也是 111, 而中間 2層個 12 個數之和則是幻和 111 兩倍 , 即222, 如圖 . 按照楊輝的數學功底 , 聚 6 圖理應有如此豐富的內涵 .
明代程大位、清代方中通、张潮、保其寿等，都曾在此基础上进一步研究纵横图。
清朝的張潮生於 1650 年 ,卒年不詳 , 著有
2 卷 .
张潮
保其壽與張潮同為清人 ,但似在張潮之後 , 因為他在“碧奈山房集”的自序中說他看過張潮的” “並指出張潮的幻方與變形幻方”所演皆平圖 ,不知立方與渾圓尤為可喜 , 其圓與洛書 , 其巧實不可思議 ,當是天地間合有此一種理數” . 因此保其壽的變形幻方別具一格 , 向立體方向發展 .
Complete bipartite graph
A graph with p vertices and q edges is graceful if there is an injective mapping f from the vertex set V(G) into{0,1,2,…,q} such that the induced map f*:E(G){1,2,…,q}
defined by f*(e)= |f(u)-f(v)| where e=(u,v), is surjective.
Graceful graph labelings were first introduced by Alex Rosa (around 1967) as means of attacking the problem of cyclically decomposing the complete graph into other graphs.
• 陈省身（ S.S. Chern ， 1903—2004 ）生辰： 1911 年 10月 28 日 祭日： 2004 年 12月 3 日 籍贯：浙江嘉兴秀水县
“ 中国数学要独立，……，从此不要再跟着人家，想法子找到新的有意义的方向，在这个方向中国数学家领头地做一些工作，不再跟了，……。中国人应该搞中国自己的数学，不要老是跟着人家走，……，中国数学应该有自己的问题，即中国数学家在中国本土上提出，而且加以解决的问题，……，我们可以根据自己的情况，挑选自己的数学研究课题，题目不必都选热门的。”
• .
Edge graceful graphs
S.P. Lo, On edge-graceful labelings of graphs, Congressus Numerantium, 50 (1985), 231-241.
Sheng-Ping Lo and Sin-Min Lee
边魔图• 边魔图的理论，是我在 1989 年与谢声忠及陈四庆教授建立的标号理论。
边魔图• 我们可以明显的看出完备图 K2 是边魔图，可是 P3却不可能是边魔图。
边魔图• 例一： 底下是 K5边魔图的一个标号。
（图）
边魔图
边魔图• 定理：除了 K2 之外，任何树 T 都不是边魔
图• 证明：任何树 T 都有两个悬挂点 u ， v ，即 d(u) = d(v) + 1 ；假设他们的边各用 x ，y来标号，则 xΞy (mod p) ；可是 x ， y ε{1,2,…p-1} ，他们的差不可能是能被 p整除的。
边魔图• 定理：任何圈 Cn ， n≥3 不是边魔图• 证明：假定 Cn 有三个邻接的边有标号，各
为 x ， y ， z 。则 x ， y相邻的点的编号是 x+y 。 y ， z相邻的点标号是 y+z 。然而我们有 x+y Ξy+z (mod p) ，即 x Ξz (mod p) 。同理这是不可能的。
边魔图• 利用类似的证明方法，我们有• 定理：如果图 G 是唯一圈图（ unicyclic gr
aph ），则 G 不是边魔图。
边魔图• 定理：完备图 Kn ，当 n Ξ0 (mod 4) 时不
是边魔图。• 证明：设 p (Kn) = 4t 则 • q (Kn) = p (p+1) / 2 = 2t (4t-1) = 8t2-2t
• 由边魔图的必要条件，我们有 q (q+1) Ξ0 (mod p)
• 即 2t (4t-1) (8t2 – 2t + 1) Ξ0 (mod 4t)
• 化简为 2t Ξ 0 (mod 4t) ，这是不可能的。
边魔图• 我们定义： Cn – 1 + K1 为轮Wn 。• 这里我们可证明定理• 轮Wn 当 n Ξ0 (mod 4) 时不会是边魔图。
On the k-Edge-Magic Complete Bipartite Graphs
Yi hui Wen Suzhou Science and Technology College
Sin-Min Lee Department of Computer Science
San Jose State University
Hugo Sun Department of Mathematics
California State University
2008.5. Shanghai
1. Introduction• Definition 1.1 Let G be a (p,q)-graph in
which the edges are labeled k, k+1,...k+q-1, where k> 0. The vertex sum for a vertex v is the sum of the labels of the incident edges at v. If the vertex sums are constant, mod p, then G is said to be k-edge-magic ( in short k-EM).
Example 1 • Figure 1 shows a graph G with 6 vertices
and 8 edges that is 1-EM with different constant sums.
• If k =1, then G is said to be edge-magic. The concept of edge-magic graphs was introduced by Lee, Seah and Tan [6].
• A necessary condition for a (p,q)-graph to be edge-magic is q(q+1)0 (mod p). However, this condition is not sufficient. There are infinitely many connected graphs such as trees and cycles satisfying this condition that are not edge-magic.
We establish the following general result:• Theorem 1.1. A necessary condition for a (p,q)-
graph to be k-edge-magic is q(q+2k-1)0 (mod p).
• Theorem 1.2. If a (p,q)-graph G is k-edge-magic
then it is pt+k-edge-magic for all t > 0 .
• A complete bipartite graph K(m,n) is the graph Nm+ Nn, where Nm is a null graph with
m vertices.
• It is obvious that K(1,n) is k-EM if and only if n=1.
2. k-Edge-magic Complete Bipartite Graphs K(2,n).
• Lemma 2.1 A necessary condition for complete bipartite graph K(s,t)to be k-edge-magic with sum c is
• 0(mod p).
0(mod p),
• Theorem 2.2. The complete bipartite graph K(2,3) is not k-edge-magic for all k .
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i
1
0
1
0
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i
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• Theorem 2.3. The complete bipartite graph K(2,4) is k-edge-magic for k1,4(mod 6) .
v1 v2 v3 v4
u1 6m+1 6m+3 6m+ 4 6m+7
u2 6m+ 2 6m+ 6 6m+ 5 6m+8
v1 v2 v3 v4
u1 6m+4 6m+6 6m+7 6m+10
u1 6m+5 6m+9 6m+8 6m+11
• Theorem 2.4. The complete bipartite graph K(2,5) is k-edge-magic for k6(mod 7) .
• Theorem 2.5. The complete bipartite graph K(2,6) is not k-edge-magic for all k .
• Theorem2.6. The complete bipartite graph K(2,7) is k-edge-magic for
k7(mod 9) .
• Theorem 2.7. The complete bipartite graph K(2,8) is k-edge-magic for
k0,5(mod 10) .• Theorem 2.8. The complete bipartite graph
K(2,9) is k-edge-magic for
k8(mod 11) .
• Theorem 2.9 The complete bipartite graph K(2,n) is not 0-edge-magic for n 10 and n 18.
• Theorem 2.10 The complete bipartite graph K(2,n) is not 1-edge-magic for n>10.
• Theorem 2.11. If n>2, K(2,n) is not 2-edge-magic .
• Theorem 2.12. If k 3, then K(2,n) is not k-edge-magic for n>8k202.
• Example 2.
The complete bipartite graphs K(2,n) is not 3 -edge-magic for all n>2 ； K(2,n) is not 4 -edge-magic for all n>10 and K(2,n) is not 5 -edge-magic for all n>18.
3. k-Edge-magic Complete Bipartite Graphs K(3,n).
• Theorem 3.1. The complete bipartite graph K(3,3) is k-edge-magic for all k .
• Theorem 3.2. The complete bipartite graph K(3,4) is k-edge-magic for all k 5(mod 7).
• Theorem 3.3. The complete bipartite graph K(3,5) is k-edge-magic for all k 1(mod 8).
• Theorem 3.4. The complete bipartite graph K(3,6) is k-edge-magic for all k.
• Theorem 3.5. The complete bipartite graph K(3,7) is k-edge-magic for all k0,5(mod 10) .
• Theorem 3.6. The complete bipartite graph K(3,8) is k-edge-magic for all k 5(mod 11).
• Theorem 3.7. The complete bipartite graph K(3,9) is k-edge-magic for all
• k1,3,5,7,9,11(mod 12).
• Theorem 3.8. The complete bipartite graph K(3,n) is not 0-edge-magic for n 10 and n88.
• Theorem 3.9. The complete bipartite graph K(3,n) is not 1-edge-magic for n 10 and n15,21,33,69.
• Theorem 3.10. The complete bipartite graph K(3,n) is not 2-edge-magic for n 10 and n15,24,51.
• Theorem 3.11. If k 3,k5, then K(3,n) is not k-edge-magic for n>18k903.
• Example 3.
The complete bipartite graphs K(3,n) is not 3-edge-magic for all n>33 ； K(3,n) is not 4 -edge-magic for all n>15,and K(3,n) is not 6 -edge-magic for all n >15.
4. . k-Edge-magic Complete Bipartite Graphs K(4,n).
• Theorem 4.1. The complete bipartite graph K(4,4) is k-edge-magic for all k .
• Theorem 4.2. The complete bipartite graph K(4,5) is k-edge-magic for all k 4(mod 9).
• Theorem 4.3. The complete bipartite graph K(4,6) is k-edge-magic for all k1,6(mod 10).
• Theorem 4.6. The complete bipartite graph K(4,9) is k-edge-magic for k2(mod 13) .
• Theorem 4.7. The complete bipartite graph K(4,n) is not 0-edge-magic for n 10 and n30,64,136,268.
• Theorem 4.8. The complete bipartite graph K(4,n) is not 1-edge-magic for n 10, and n11,12,16,20,26,36,44,56,76,116,236.
• Theorem 4.9. The complete bipartite graph K(4,n) is not 2-edge-magic for n 10 and n12,22,48,100,204.
• Theorem 4.10. If k 3, then K(4,n) is not k-edge-magic for n>8(344k)4.
• Example 4.
• The complete bipartite graphs K(4,n) is not 3-edge-magic for all n>172 ； K(4,n) is not 4 -edge-magic for all n>140,and K(4,n) is not 6 -edge-magic for all n >108.
5.General results.
• Theorem 5.1. The complete bipartite graph K(n,n) is k-edge-magic for all k ,when n >4.
• Theorem 5.2
The complete bipartite graph K(n,2n) is 0-edge-magic ,for n=0(mod3).
• Corollary 5.3. The complete bipartite graph K(n,2n) is k-edge-magic for all k ,and n=0(mod3).
• Example 5.
The complete bipartite graph K(6,12) is k-edge-magic for all k.
• Theorem 5.4.
The complete bipartite graph K(n,3n) is not k-edge-magic for all k, when n=2(mod4).
• Example 6. The complete bipartite graphs K(6,18), K(10,30) and K(14,42) are not k-edge-magic for all k .
6.Conjecture.• Conjecture.
A necessary and sufficient condition for a (p,q)-graph to be k-edge-magic is q(q+2k-1)0 (mod p).
References • [1] Dharam Chopra, Rose Dios and S. M. Lee, On the edge-magicn
ess of Zykov sum of graphs , to appear in JCMCC..• [2] Dharam Chopra, H. Kwong and Sin-Min Lee, On The Edge-magi
c (p,3p-1)-Graphs, Congressus Numerantium 179, 49-63, 2006• [3]J.A. Gallian, A dynamic survey of graph labeling, The Electronic J.
of Combin. (2007), # DS6, 1-180.• [4] N. Hartsfield and G. Ringel, Supermagic and Antimagic Graphs,
Journal of Recreational Mathematics, 21 ,107-115, 1989.• [5] Y.S. Ho and Sin-Min Lee, An initial result of supermagicness of c
omplete k-partite graphs, Journal of Combinatorial Mathematics and Combinatorial Computing, 39 (2001) 3 - 17
• [6] Sin-Min Lee, W.M. Pigg and T.J. Cox, On edge-magic cubic graphs conjecture, Congressus Numerantium, 105 ,214-222, 1994.
• [7] Sin-Min Lee, Eric Seah and S.K. Tan, On edge-magic graphs, Congressus Numerantium, 86, 179-191, 1992.
• [8] Sin-Min Lee, E. Seah, Siu-Ming Tong, On the edge-magic and edge-graceful total graphs conjecture, Congressus Numerantium 141, 37-48 ,1999
• [9] Sin-Min Lee, Ling.Wang and Yihui Wen, On The Edge-magic Cubic Graphs and Multigraph, Congressus Numerantium 165 ,145 – 160, 2003.
• [10] Joseph Needham, Sciences and Civilization in China III, Cambridge University Press, 56-62, 1969.
• [11] Karl Schaffer and Sin Min Lee, Edge-graceful and edge-magic labellings of Cartesian products of graphs, Congressus Numerantium 141, 119-134, 1999.
• [12] W.C. Shiu, P.C.B. Lam and H.L. Cheng, Supermagic labeling of sKn,n, Congressus Numerantium, 146,119-124, 2000.
• [13] W.C. Shiu, P.C.B. Lam and Sin-Min Lee, On a construction of supermagic graphs, Journal of Combinatorial Mathematics and Combinatorial Computing 42,147-160,2002.
• [14] W.C. Shiu, P.C.B. Lam and Sin-Min Lee, Edge-magicness of the composition of a cycle with a null graph, Congressus Numerantium, 132 ,9-18, 1998.
• [15] W.C. Shiu, F.C.B. Lam and Sin-Min Lee, Edge - magic index sets of (p, p-1)-graphs, Electronic Notes in Discret Mathematics, Vol. 11 , 2002.
• [16] W.C. Shiu and Sin-Min Lee, Some edge-magic cubic graphs, Journal of Combinatorial Mathematics an Combinatorial Computing, 40 , 115-127, 2002.
• [17] Y.H. Wen, Sin-Min Lee and Hugo Sun, On supermagic edge splitting extension of graphs, Ars Combinatoria 79, 115-128 2006.
Thanks!
On The Integer Magic Spectra of the Eulerian Graphs of
Odd Sizes
Sin-Min Lee and Richard LowSan Jose State University
Kam Chuen NgEastman Kodak Company
A-magic Graph
• For any abelian group written additively we denote
• Any mapping l: E(G) is called a labeling Given a labeling on edge set of G we can induce a vertex set labeling l+: V(G) as follows l+(v)= {l(u,v) : (u,v) in E(G)}
• A graph G is called an -magic if there is a labeling l: E(G) such that for each vertex v, the sum of the labels of the edges incident with v are all equal to the same constant; i.e., l+(v) = c for some fixed c in A. We will called <G,l> a A-magic graph with sum c.
We denote the class of all graphs (either simple or multiple graphs) by Gph.
The class of all abelian groups by Ab.
For each in Ab we denote the class of all -magic graphs by Gp.
We call a graph G fully magic if it is in Gp for all non-trivial abelian group A.
c1
c5
c4
x
-x
x
-x
c2
-x
x x
-x
v1
v2 v3
c3
C(3,5) is fully magic C(4,6) is fully magic
c1
c5
c4
x
-x x
-x
c2-x
x
x-x
v1
v2
v3
c3
-x
v4
xc6
When A = Z, the Z-magic graphs were considered in Stanley.
He pointed out that the theory of magic labelings can be put into the more general context of linear homogeneous diophantine equations.
When the group is Zk, we shall refer to the Zk –magic graph as k-magic.
R.M. Low and S-M. Lee, On group-magic eulerian graphs, J. Combin, Math. Combin. Computing, 50 (2004), 141-148.
Theorem 1.1. Every eulerian graph G is k-magic, for k even.
Theorem 1.2. If G is an eulerian graph of even size, then G is fully magic.
For any cycle Cn, and S E(Cn) with a mapping f: S N\ {1}.
We define a multigraph
Spext(Cn,S,f) on Cn as follows:
V(Spext(Cn,S,f)) = V(Cn)
E(Spext(Cn,S,f)) = { e1,e2,…,et : if f(e) = t} {f : f
E(Cn) \ S}.
We call Spext(Cn,S,f) the split extension of (Cn,S,f).
Family 1. Let n =2s. S = {(v4,v5), (v6,v7),…(v2s,v1)} and f:SN
has the property
f(e) is odd for each e in S.
Theorem 2.1. IM(Spext(C2s,S,f)) = N.
Family 2. Let n =2s+1. S = {(v3,v4), (v5,v6),…(v2s+1,v1)} and
f:SN has the property
f(e) is odd for each e in S.
Use the similar argument as Theorem 2.1., we can show that
Theorem 2.2. IM(Spext(C2s+1,S,f)) = N.
Family 3. For any cycle Cn , with a mapping f: E(Cn) N\ {1}.
We define a multigraph
Spext(Cn,S,f) on Cn as follows:
V(Spext(Cn, E(Cn),f)) = V(Cn)
E(Spext(Cn, E(Cn),f)) = { e1,e2,…,et : if f(e) = t}
Theorem 2.3. If n is even and f: E(Cn) N\ {1} with the property
|f-1()| is odd, then IM(Spext(Cn, E(Cn),f))= N.
Family 4.
Theorem 2.4. IM(R(2s,2t)) = N.
Family 6. B(m,n)
Theorem 2.6. IM(B(2t,2s+1)) = N for all t > 2 and s > 1.
B(4,5) is 2k+1-magic.
Family 7. T(m,n)
Given a triangle with vertex set {c1,c2,c3}, and integers
m, n > 3. The graph T(m,n) is constructed by amalgamate c1 with a cycle Cm, and c3 with a cycle Cn. It is obvious that
T(m,n) is an eulerian graph.
Theorem 2.7. IM(T(m,n)) = N , if m+n is even.
Integer-magic spectra of the eulerian graphs which are not N.
Family 1.
Theorem 3.1.. If n is odd and f: E(Cn) N\ {1} with the prop
erty |f-1()| is odd, then IM(Spext(Cn, E(Cn),f))= 2N.
Family 2.
Recall a two cycle C2 is a (2,2) - graph with two vertices and tw
o parallel edges.
For any cycle Cn, and a mapping f: E(Cn) N\ {1}. We define
a multigraph
Aext(Cn ,f) on Cn as follows:
For each e E(Cn) with f(e) = t, then we attach a cycle Ct in Cn.
Theorem 3.2.. If f: E(Cn) N\ {1} with the property |f-1()| is o
dd, then IM(Aext(Cn, f)) = 2N
IM(Aext(C4, f)) = 2N
Family 3. (2k,2s)
Let X = {x1,x2} and Y = {y1,y2,…,y2k} where X is P2 and Y is
null graph N2k. We first form X+Y. Then we amalgamate X+Y
with a path of length 2s by identified its two end vertices with x1 and x2 respectively (see Figure 12 for k=2 and s =2). We denote the final graph by (2k,2s).
Theorem 3.3. The graph (2k,2s) has integer magic spectrum 2N for any k > 1 and s > 1.
c5
c4
t+1
t
1
c2
t
t t
t+1
v1
v2 v3
c3
C(3,5) is 2t+1-magic
c1
t
t+1c5
c4
t+1
t-1
2
c2
t-1
t-1 t-1
t+1
v1
v2 v3
c3
C(3,5) is 2t-magic
c1
t-1
t+1
Let N be the set {1,2,3,…} and for each k>0, we write the set {kx: x in N} by kN.
For simplicity, we will consider Z-magic as 1-magic.
Given a graph G, we denote the set of all k> 0 such that G is k-magic by IM(G). We call this set as integer-magic spectrum of G.
C(m,n) double cycle.
Theorem 1. If m+n is even, then IM(C(m,n)) = N.
Family 4. (n,g)
For any cycle Cn, and a mapping g: V(Cn) N\ {1}. We de
fine a multigraph
Bext(Cn ,g) on Cn as follows:
For each v V(Cn) with g(v) = t, then we amalgamate a cyc
le Ct on v. We denote the final graph by (n,g).
Let g: C5 N\{1} defined
by g(c1)= 4, g(c2) = 3, g(c3)
= 2, g(c4) = 6, g(c5 )=3
c1
c4
c2
c3
(4,g) is 2k+1-magic
k
k
k
k
k k
k kk
k k
kk
k
k
k+1
k+1
Theorem 3.4. If n is even > 4 and with the property |g-1()| is odd, the graph (n,g) has integer magic spectrum N \{1}.
c1
c4
c2
c3
c5
Let g: C5 N\{1} defined by g(c1)= 4, g(c2) = 3, g(c3) = 2,
g(c4) = 6, g(c5 )=3. IM( (n,g)) = 2N.
Theorem 3.5. If n is odd > 3, and with the property |g-1()| is even. the graph (n,g) has integer magic spectrum 2N .
