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Experts and Learners Good afternoon

Experts and Learners Good afternoon. i. Back. Close. On the edge-balance index sets of Chain Sums of K 4 -e. Yu-ge Zheng* Juan Lu* Sin-Min Lee**. * (Department of Mathematics, Henan Polytechinc University Jiaozuo 454000,P.R.China)

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Experts and Learners Good afternoon

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  1. Experts and Learners Good afternoon i Back Close

  2. On the edge-balance index sets of Chain Sums ofK4-e Yu-ge Zheng* Juan Lu* Sin-Min Lee** • *(Department of Mathematics, Henan Polytechinc University Jiaozuo 454000,P.R.China) • * (Department of Computer Science San Jose State Univer- • sity San Jose, CA95192,USA) Back Close

  3. Introduction Ⅰ Ⅱ On the edge-balance index sets of the first type of chain-sums of Back Ⅲ On the edge-balance index sets of the second type of chain-sums of Close

  4. Introduction Use the vertex and edge of graph labeling function theory to study the graphs by B.M. Stewart introduced in 1966, over the years, many domestic and foreign re- searchers to work closely with the research in this area, and accessed to a series of research results, such as gra- phical construction method of edge-magic graphs and edge-graceful graphs, their theoretical study and so on. Back Close

  5. Introduction Boolean index sets of graphs are that make the vertex sets and the edge sets of graphs through the mapping function with corresponding, to study the charac- teristics of various types of graphs and the inherent characteristics of graphs, Boolean index set theory can be applied to information engineering, commu- unication networks、computer science、economic management、medicine, etc. The edge-balance index set is an important issue in Boolean index set . Back Close

  6. Introduction In this paper, we will introduce the edge-balance index sets of chain sums of mainly. Let G be a graph with vertex set V(G) and edge set E(G), and let Z2={0, 1}. A labelingf : E(G)→Z2 induces a partial vertex labeling f+ : V(G) → Z2 defined byf+ (x) = 0 if the edge labeling of f(xy) is 0 more than 1and f+ (x) = 1 if the edge labeling of f(xy) is 1 more than 0. Back Close

  7. Introduction f+ (x) is not defined if the number of edge labeling with 0 is equal to the number of edge labeling with 1. For i Z2, vf (i) = v (i) = card{v V(G) : f+ (V) = i} and ef (i) = e (i) = card{e E(G) : f (e) = i}. Back Close

  8. Main content Definition 1. A labeling f of a graph G is said to be edge-friendly if Definition 2. The edge-balance index set of the graph G, EBI(G), is defined as {| |: the edge labeling f is edge-friendly } Back Close

  9. Main content Definition 3. Let G be a graph and u, v be two distinct vertices of G. We construct the graph Pn(G, {u, v}) as follows. The vertex set V(Pn(G, {u, v})) is the union of (n-1) copies of V(G). We denote the vertices u, v in the copy of V(G) by ui, vi. The edge set E(Pn(G, {u, v})) is the union of (n-1) copies of E(G) with vj and uj-1 ident- ifined for j=1,2,…,n-2.We call Pn (G, {u, v}) the chain sum of (G, {u, v}) by Pn. Back Close

  10. Main content For n > 3, we denote Pn(K4-e, {x, z}) by B(n-1), B(n-1) is said to be the first type of chain-sums of K4-e else; we denote Pn(K4-e, {u, v}) by H(n-1), H(n-1) is said to be the second type of chain-sums of K4-e. As shown in Figure 1. Back Close

  11. Main content Definition 4. Let G be a continuous subgraph of graph H(n) (or B(n)), and the number of 1-edge be m in G. If the difference between the number of 1-vertices and the number of 0-vertices is the largest in graph G, then the graph G is said to be the best m-edge graph. (We say an edge labeling of f(xy) is 1 to be 1-edge and an edge labeling of f(xy) is 0 to be 0-edge, an vertex labeling is 1 to be 1-vertex and an vertex labeling is 0 to be 0-vertex when the context is clear. ) Back Close

  12. Main content Definition 5. The complete index set of B(n), CEBI(B(n)) is defined as EBI(B(n)) = {0, 1, … , k}. Definition 6. The perfect index set of B(n), PEBI(B(n)), is defined as CEBI(B(n)) = {0+3i, 1+3i, 2+3i, i=0, 1, … , }. Back Close

  13. Main content 一、 On the edge-balance index sets of the first type of chain-sums of K4-e Lemma 1. if EBI(B(2n-1)) = {0, 1, … , k}, then EBI (B(2n+1)) includes {0, 1, … , k} Back Lemma 2. The perfect index set of B(1) exists, and PEBI(B(1)) = {0, 1, 2}. Close

  14. Main content Lemma 3. The perfect index set of B(2) exists, and PEBI(B(2)) = {0, 1, 2}. Lemma 4. The perfect index set of B(3) exists, and PEBI(B(3)) = {0, 1, 2, 3, 4, 5}. Back Lemma 5. The perfect index set of B(4) exists, and PEBI(B(4)) = {0, 1, 2, 3, 4, 5}. Close

  15. Main content Lemma 6. The perfect index set of B(5) exists, and PEBI(B(5)) = {0, 1, 2, 3,4, 5, 6, 7, 8}. Lemma 7. The complete index set of B(6) exists but it’s perfect index set doesn’t exist, and CEBI(B(6)) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Back Lemma 8. The perfect index set of B(7) exists, and PEBI(B(7)) = {0, 1, 2, … , 11}. Close

  16. Main content Lemma 9. The perfect index set of B(8) exists, and PEBI(B(8)) = {0, 1, 2, … , 11}. Lemma 10. The perfect index set of B(9) exists, and PEBI(B(8)) = {0, 1, 2, … , 14}. Lemma 11. The perfect index set of B(10) exists, and PEBI(B(10)) = {0, 1, 2, … , 14}. Back Close

  17. Main content Lemma 12. The complete index set of B(11) exists but it’s perfect index set doesn’t exist, and CEBI(B(11)) = {0, 1, 2, … , 16}. Lemma 13. The perfect index set of B(12) exists, and PEBI(B(12)) = {0, 1, 2, … , 17}. Back Lemma 14. The perfect index set of B(13) exists, and PEBI(B(12)) = {0, 1, 2, … , 20}. Close

  18. Main content Theorem 1. The complete index set of B(n) all exist (n = 1, 2, … , 13). Theorem 2. Except B(6) and B(11), the perfect index set of B(n) all exist (n = 1, 2, 3, 4, 5, 7, 8, 9, 10, 12, 13). Theorem 3. The complete index set of B(n) all exist. Back Close Theorem 4. The perfect index set of B(n) (n is odd) exist if and only if n = 1, 3, 5, 7, 9, 13.

  19. Main content 二、 On the edge-balance index sets of the second type of chain-sums of K4-e Lemma 1 If m1 ≡ 0(mod 4) (m1 = 4k (k N)), then max{ EBI(H(n)) } = 6k-3n+1. Back Lemma 2 If m1≡ 1(mod 4) (m1= 4k+1 (k N)), then max{ EBI(H(n)) } = 6k-3n+2. Close

  20. Main content Lemma 3 If m1≡ 2(mod 4) (m1= 4k+2 (k N))), then max{ EBI( H(n) ) } = 6k-3n+3. Lemma 4 If m1 ≡ 3(mod 4) (m1 = 4k+3 (k N)), then max{ EBI(H(n)) } = 6k-3n+5. Theorem 1 If m1≡ 0(mod 4) (m1 = 4k (k N)), then EBI(H(n)) = {0, 1, 2, 3, … , 6k-3n+1}. Back Close

  21. Main content Theorem 2 If m1≡ 1(mod 4) (m1= 4k+1 (k N)), then EBI(H(n)) = {0, 1, 2, 3, … , 6k-3n+2}. Theorem 3 If m1≡ 2(mod 4) (m1= 4k+2 (k N)), then EBI(H(n)) = {0, 1, 2, 3, … , 6k-3n+3}. Theorem 4 If m1 ≡ 3(mod4) (m1= 4k+3 (k N)), then EBI(H(n)) = {0, 1, 2, 3, …, 6k-3n+5}. Back Close

  22. Main content Theorem 5 max{ EBI(H(n)) } = 3k+2 (k N) if and only if n = 4k+1 and n = 4k+2. Theorem 6 max{ EBI(H(n)) } = 2(k+t)+4 if and only if n = 4k+3 (one to one correspondence between k and t, k = 0, 1, 2, 3, … , t = 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, … , in other words, t = 0 when k = 0, t = 0 when k = 1, followed by analogy). Back Close

  23. Main content Theorem 7 max{ EBI(H(n)) } = 2(k+t)+3 if and only if n = 4k+4 (k = 0, 1, 2, 3, … , t = 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, … , one to one correspondence between k and t) Back Close

  24. Reference • [1] Bor-Liang Chen, Kuo-Ching Huang, Sin-Min Lee and Shi-Shan Liu, On edge-balanced multigraphs, Journal of Combinatorial Mathematics and Combinatorial Computing, 42(2002), 177-185 . • [2] Ebrahim Salehi and Sin-Min Lee, Friendly index sets of trees, Congressus Numerantium 178 (2006), pp. 173-183. • [3] Alexander Nien-Tsu Lee, Sin-Min Lee and Ho Kuen Ng, On The Balance Index Set of Graphs, • Journal of Combinatorial Mathematics and Combinatorial Computing.,66 (2008) 135-150. • [4] Harris Kwong, Sin-Min Lee and D.G. Sarvate, On the Balance Index Sets of One-point Unions of Graphs, Journal of Combinatorial Mathematics and Combinatorial Computing.,66 (2008) 113-127. • [5] Harris Kwong, Sin-Min Lee and H.K. Ng ,On Friendly Index Sets of 2-regular graphs, Discrete Mathematics.,308 (2008) 5522-5532. • [6] Suh-Ryung Kim, Sin-Min Lee and Ho Kuen Ng, On Balancedness of Some Graph Constructions, Journal of Combinatorial Mathematics and Combinatorial Computing.,66 (2008) 3-16. Back Close

  25. Thank you Back Close

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