1. On Balanced Index Sets of Disjoint Union Graphs Sin-Min Lee Department of Computer Science San Jose State University San Jose, CA 95192, USA Hsin-Hao Su Department of Mathematics Stonehill College Easton, MA 02357, USA Yung-Chin Wang * Department of Physical Therapy Tzu-Hui Institute of Technology Taiwan, Republic of China 40th SICCGC March 2-6, 2009

2. Definition ( A. Liu, S.K. Tan and S.M. Lee 1992 ) Let G be a graph with vertex set V(G) and edge set E(G). A vertex labeling of G is a mapping f from V(G) into the set {0, 1}. For each vertex labeling f of G, define a partial edge labeling f* of G from E(G) into the set {0, 1} as following. For each edge (u, v)  E(G), where u, v  V(G), ┌ 0, if f(u) = f(v) = 0, f*(u,v) = ┤ 1, if f(u) = f(v) = 1, └ undefined, if f(u) ≠ f(v).

3. Definition ( A. Liu, S.K. Tan and S.M. Lee 1992 ) A graph G is said to be a balanced graph or G is balanced if there is a vertex labeling f of G satisfying |v f (0) – v f (1)| ≤ 1 and |e f* (0) – e f* (1) | ≤ 1.

4. Definition ( A.N.T. Lee, S.M. Lee, H.K. Ng 2008 ) The balance index set of a graph G, BI(G), is defined as {|e f* (0) – e f* (1)| : the vertex labeling f is friendly}.

5. Example. BI(K 3,3 ) = {0}

6. Example. BI(DS(2,2)) = {0,2}, BI(DS(3,3)) = {0,3}.

7. Theorem (Kwong, Lee, Lo, Wang 2008) Let G be a k-regular graph G of order p. Then ┌{0} if p is even, BI(G) =┤ └{k/2} if p is odd.

8. Permutation Graphs Let  be a permutation of the set [n]= {1,2,…,n}. For a graph G of order n, the  -permutation graph of G is the disjoint union of two copies of G, namely, G T and G B, together with the edges joining the vertex v i of G T with v  (i) of G B.

9. Theorem (Lee & Su) Let G and H be two graphs with the same number of vertices and G ∪ H be the disjoint union of these two graphs. Let  be any permutation between the vertex sets of G and H. Then, the balance index set BI(Perm(G, ,H)) = BI(G ∪ H).

10. Theorem (Lee & Su) Let G and H be two graphs with the same order, if both of them are k-regular graphs, then BI(G ∪ H)={0}.

11. Example Let G and H be two 4-regular graphs as below, then BI(G ∪ H)={0}.

12. Lemma Let f be a friendly labeling of the disjoint union G ∪ H of two graphs G and H, where G and H have the same number of vertices. Then, the number of 0-vertices of G equals the number of 1-vertices of H and the number of 1-vertices of G equals the number of 0-vertices of H, i.e., v G (1) = v H (0) and v G (0) = v H (1).

13. Theorem For any G in REG(s) and H in REG(t) of order n and any friendly labeling f on G ∪ H, we have 2( e(0) - e(1) ) = ( s - t )( v G (0) - v H (0) ) = ( s - t )( 2v G (0) - n ) = ( s - t )( n - 2v H (0) )

14. Theorem Let G and H be two graphs with the same order n, if G is a k-regular graph and H is an h-regular graph, k≠h, then 1.{ 0, |s-t|, 2|s-t|, 3|s-t|, …, (n/2)|s-t| }, if n is even, 2.{ |(s-t)/2|, 3|(s-t)/2|, 5|(s-t)/2|, …, n|(s-t)/2| }, if n is odd.

15. Example BI(C 4 ∪ K 4 )={0,1,2}

16. Theorem BI(C n ∪ P n ))={0,1}. Example. BI(C 6 ∪ P 6 )={0,1}

17. Theorem BI(C n ∪ St(n-1))={0,1,2,…,n-2}. Example. BI(C 4 ∪ St(3))={0,1,2}

18. Theorem BI(P n ∪ St(n-1))={0,1,2,…,n-2} Example. BI(P 6 ∪ St(5))={0,1,2,3,4}

19. Theorem. Let BI(SP(2 [n] )) be the spider. We have 1.BI(SP(2 [n] )) = {0,1,…,n} 2.BI(SP(2 [n] ) ∪ SP (2 [n] ) )={0,1,2,…,2(n-1)} SP(2 [3] )

20. Theorem. Let CT(1 [n] ) be the corona of a path P n. We have 1.BI(CT(1 [n] ) )={0,1,2,…,n-1} 2.BI(CT(1 [n] ) ∪ CT(1 [n] ) )={0,1,2,…,2(n-1)} CT(1 [5] )

21. Theorem Let DS(m, n) be the double star. We have 1.{(n – m)/2, (n + m)/2}, if m + n is even, 2.{(n – m – 1)/2, (n – m + 1)/2, (n + m – 1)/2, (n + m + 1)/2}, if m + n is odd.

22. Unsolved Problem For what m,n, BI(DS(m,n) )  DS(m,n))) forms arithmetic progression?