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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. Yun g-Chin Wang * Department of Physical Therapy
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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
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).
Definition (A. Liu, S.K. Tan and S.M. Lee 1992) A graph G is said to be a balancedgraph or G is balanced if there is a vertex labeling f of G satisfying |vf(0) – vf(1)| ≤ 1 and |ef*(0) – ef*(1) | ≤ 1.
Definition (A.N.T. Lee, S.M. Lee, H.K. Ng2008) The balance index set of a graph G, BI(G), is defined as {|ef*(0) – ef*(1)| : the vertex labeling f is friendly}.
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.
Permutation Graphs Let be a permutation of the set [n]= {1,2,…,n}.For a graph G of order n,the -permutation graph of Gis the disjoint union of two copies of G, namely, GT and GB, together with the edges joining the vertex vi of GT with v(i) of GB.
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).
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}.
Example Let G and H be two 4-regular graphs as below, then BI(G ∪ H)={0}.
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., vG(1) = vH(0) and vG(0) = vH(1).
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 )( vG(0) - vH(0) ) = ( s - t )( 2vG(0) - n ) = ( s - t )( n - 2vH(0) )
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 • { 0, |s-t|, 2|s-t|, 3|s-t|, …, (n/2)|s-t| }, if n is even, • { |(s-t)/2|, 3|(s-t)/2|, 5|(s-t)/2|, …, n|(s-t)/2| }, if n is odd.
Example BI(C4 ∪ K4)={0,1,2}
Theorem BI(Cn∪Pn))={0,1}. Example. BI(C6∪P6)={0,1}
Theorem BI(Cn∪St(n-1))={0,1,2,…,n-2}. Example. BI(C4∪St(3))={0,1,2}
Theorem BI(Pn∪St(n-1))={0,1,2,…,n-2} Example. BI(P6∪St(5))={0,1,2,3,4}
Theorem. Let BI(SP(2[n])) be the spider. We have • BI(SP(2[n])) = {0,1,…,n} • BI(SP(2[n]) ∪ SP (2[n]) )={0,1,2,…,2(n-1)} SP(2[3])
Theorem. Let CT(1[n]) be the corona of a path Pn. We have • BI(CT(1[n]) )={0,1,2,…,n-1} • BI(CT(1[n]) ∪ CT(1[n]) )={0,1,2,…,2(n-1)} CT(1[5])
Theorem Let DS(m, n) be the double star. We have • {(n – m)/2, (n + m)/2}, if m + n is even, • {(n – m – 1)/2, (n – m + 1)/2, (n + m – 1)/2, (n + m + 1)/2}, if m + n is odd.
Unsolved Problem For what m,n, BI(DS(m,n)) DS(m,n))) forms arithmetic progression?