On Balanced Index Sets of Disjoint Union Graphs

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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
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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) .

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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 vf(0) vf(1) 1 and ef(0)
  ef(1) 1.

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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 ef(0) ef(1) the vertex
  labeling f is friendly.

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Example. BI(K3,3) 0
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Example. BI(DS(2,2)) 0,2,
BI(DS(3,3)) 0,3.
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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.

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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, GT and GB, together with
  the edges joining the vertex vi of GT with v?(i)
  of GB.

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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).

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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.

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Example

• Let G and H be two 4-regular graphs as below,
• then BI(G ? H)0.

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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).

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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) )

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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, 2s-t, 3s-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.

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Example

• BI(C4 ? K4)0,1,2

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Theorem

• BI(Cn?Pn))0,1.

Example. BI(C6?P6)0,1
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Theorem

• BI(Cn?St(n-1))0,1,2,,n-2.

Example. BI(C4?St(3))0,1,2
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Theorem

• BI(Pn?St(n-1))0,1,2,,n-2

Example. BI(P6?St(5))0,1,2,3,4

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Theorem.

• Let BI(SP(2n)) be the spider.
• We have
• BI(SP(2n)) 0,1,,n
• BI(SP(2n) ? SP (2n) )0,1,2,,2(n-1)

SP(23)
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Theorem.

• Let CT(1n) be the corona of a path Pn.
• We have
• BI(CT(1n) )0,1,2,,n-1
• BI(CT(1n) ? CT(1n) )0,1,2,,2(n-1)

CT(15)
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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.

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Unsolved Problem

• For what m,n, BI(DS(m,n) ) ? DS(m,n))) forms
  arithmetic progression?