Int. Journal of Math. Analysis, Vol. 6, 2012, no. 40, 1997 - 2005
Harmonic Mean Labeling of Some
Cycle Related Graphs
S. S. Sandhya
Department of Mathematics, SreeAyyappa College for Women
Chunkankadi, Kanyakumari 629 807, Tamilnadu, India
sssandhya2009@gmail.com
S.Somasundaram
Department of Mathematics, ManonmaniamSundaranar University
Tirunelveli 627 012,Tamilnadu, India
somumsu@rediffmail.com
R.Ponraj
Department of Mathematics, Sri Paramakalyani College
Alwarkurichi 627 412, Tamilnadu, India
ponrajmaths @indiatimes.com
Abstract
In this paper we discuss Harmonic mean labeling behaviour of some cycle related
graphs such as duplication, joint sum of the cycle and identification of cycle. Also we
investigate Harmonic mean labeling behaviour of Alternate Triangular snake A(Tn),
Alternate Quadrilateral snake A(Qn).
Keywords: Harmonic mean labeling, Duplication, Joint sum, Identification, Alternate
Triangular snake, Alternate Quadrilateral snake
1. Introduction
The graph considered here will be finite, undirected and simple graph G= (V,E)
with p vertices and q edges. For a detailed survey of graph labeling we refer to Gallian
[1]. Terms not defined here are used in the sense of Harary [2].
S. S. Sandhya, S. Somasundaram and R. Ponraj
1998
We shall make frequent references to the following definitions
Definition 1.1: A graph G= (V,E) with p vertices and q edges is called a Harmonic mean
graph if it is possible to label the vertices x ∈V with distinct lables from 1,2…q+1 in
ଶ௙ሺ௨ሻ௙ሺ௩ሻ
ଶ௙ሺ௨ሻ௙ሺ௩ሻ
such a way that when each edge e=uv is labeled with f(e=uv) = ቒ
ቓ(or) ቔ
ቕ
௙ሺ௨ሻା௙ሺ௩ሻ
௙ሺ௨ሻା௙ሺ௩ሻ
then the edge labels are distinct. In this case, f is called a Harmonic mean labeling of G.
Definition 1.2[10]:Let v be a vertex of a graph G. Then the duplication of v is a graph
G(v) obtained from G by adding a new vertex v’with N(v′)=N(v).
Definition 1.3 [10]:Let e=uv be an edge of G. Then duplication of an edge e=uvis a
graph G(uv) obtained from G by adding a new edge u ′v ′ such that
N(u′)=N(u) ∪{v′}-{v}and N(v′)= N(v)∪{u′}-{u}
Definition 1.4 [8]: Consider two copies of Cn, connect a vertex of first coy to a vertex of
second copy with a new edge, the new graph obtained is called joint sum of Cn.
Definition 1.5 [9]: Let u and v be two distinct vertices of a graph G. A new graph G1 is
constructed by identifying two vertices u and v by a single vertex w is such that every
edge which was incident with either u or v in G is not incident with w in G.
The notion of Harmonic mean labeling was introduced by S.Somasundaram
S.S.Sandhya and R.Ponraj and the Harmonic mean labeling behaviour of Path Pn, Cycle
Cn, Star K1,n, n≤7, Complete graph Kn, n≤3, Triangular snake Tn Quadrilateral snake Qn,
Comb, Ladder has been investigated in [3] and [4]. In this paper we contribute some new
results for Harmonic mean labeling graphs.
2. Main Results
Theorem 2.1: The graph obtained by duplicating an arbitrary vertex of Cn admits a
Harmonic mean labeling.
Proof:
Let Cn = v1v2…….vnv1 be the cycle
Let vi′ be the duplicated vertex of vi.
Define a function f: V(G(vi)) →{1,2…..q+1}
by f(v1) = n+3, f(v1′) =1
f(vi) = i+3, 2 ≤i≤n
Hence f is harmonic mean labeling of the graph G(vi).
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Example 2.2: A Harmonic mean labeling C5(v) is shown in figure 1.
8
Figure: 1
In the similar manner we have the following
Theorem 2.3: The graph obtained by duplicating an arbitrary edge in cycle Cn is a
Harmonic mean graph.
Proof:Let Cn = v1 v2….vn v1 be the cycle.
Let e′= u1′u2′ be the duplicated edge of e = u1 u2
Now we define f: V(G(u1u2)) →{1,2….q+1}
by f(u1) = 1, f(u2) = 2 , f(u3)=3
f(ui) = i+1, 4≤i≤n
f(u1′) =n, f(u2′) = n+1
Hence f is a Harmonic mean labeling of duplicated graph G(u1u2).
Example 2.4: The following is the harmonic mean labeling of C9(u1u2)
S. S. Sandhya, S. Somasundaram and R. Ponraj
2000
Figure :2
Next we have
Theorem 2.5: The joint sum of two copies of Cn admits a Harmonic mean labeling
Proof: Let G be the joint sum of the cycle Cn
Let v1 v2….vnv1 be the first copy of Cn and vn+1vn+2…v2nvn+1 be the second copy of Cn.
Without loss of generality we can assume that vn and vn+1 are joined in G.
Now define a function f: V(G) →{1,2…q+1} by
f(vi) =
i,1 ≤ i ≤ n
n+1, i = n+1
i+1, n+2 ≤ i ≤ 2n
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Clearly f is a Harmonic mean labeling.
Example 2.6: Joint sum of C7 and its Harmonic mean labeling is shown in figure 3.
1
Figure: 3
Now we prove the following
Theorem 2.7: The graph Cn(2) is a Harmonic mean graph
Proof:Let u be the central vertex of Cn(2). Let u1u2…….unu1 and v1v2….vnv1be the
vertices of first and second cycle of Cn(2) respectively.
take u = un = v1
Define f:V (Cn(2)) →{1,2….q+1} by
f(u) = n+1
2002
S. S. Sandhya, S. Somasundaram and R. Ponraj
f(ui)=2, 1≤i≤n
f(vi)=n+1+i, 2≤i≤n-1
f(vm) = 2m
Obviously f is a Harmonic mean labeling of Cn(2)
Example 2.8: A harmonic mean labeling of C7(2) is shown below
Figure:4
Next we have
Definition 2.8: An Alternate Triangular snake A(Tn) is obtained from a path u1u2…..un by
joining ui and ui+1 (alternatively) to new vertex vi
That is every alternate edge if a path is replaced by C3
Theorem 2.9: Alternate Triangular snakes A(Tn) are Harmonic mean graphs
Proof: Here we consider the following two cases
Case (i): If the triangle starts from u2.
Define a function f: V A(Tn) →{1,2….q+1}
by
f(u1) = 1
f(u2)=2
f(ui) = 2i-2 for all i=3,4…n
f(vi) -=2i-1 for all i=2 ,4…n-2
The edges are labeled with
f(uiui+1) = 2i-1 for all i=1, 2……n-1
f(uivi) = 2i-2 for all i=2,4…n-2
f(viui+1)= 2i for all i= 2,4,6…n-2
In this case f is a harmonic mean labeling of A(Tn)
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Figure : 5
Case(ii): If the triangle starts for u1.
Define a function f : V A(Tn)→{1,2….q+1}
by
f(ui) = 2i-1 for all i=1,2…n
f(vi) = 2i for all i=1,3….n-1
The edges are labeled with
f(uiui+1) = 2i for all i=1,2….n
f(uivi)=2i-1 for all i=1,3,5…n-1
f(vivi+1) = 2i+1 for all i=1,3,5….n-1
The f is a harmonic mean labeling of A(Tn)
Figure : 6
From case (i) and case (ii) we conclude that Alternate Triangular snake is a Harmonic
mean graph.
In the same manner we have
Definition 2.10: An Alternate Quadrilateral snake A(Qn) is obtained from a path
u1u2…..un by joining ui, ui+1 (alternatively) to new vertices viwi respectively and then
joining vi and wi.
That is every alternate edge of a path is replaced by a cycle C4
Now we prove the following
2004
S. S. Sandhya, S. Somasundaram and R. Ponraj
Theorem 2.11: Alternate Quadrilateral snakes A(Qn) are Harmonic mean graphs.
Proof: Let AQn be the Alternate Quadrilateral snake
Here we consider two different cases
Case(i): If the Quadrilateral starts from u2.
Define a function f: V(AQn)→{1,2…q+1}
by f(u1) = 1, f(u2)=2, f(u3)=5
f(ui) = f(ui-2)+5 for all i=4 , 5 …n-1
f(vi)=3, f(v2)=4
f(vi)=f(vi-2)+5 for all i=3,4…n-1
Edges are labeled with
f(u1u2)=1, f(u2u3)=3
f(uiui+1) = f(ui-2 ui-1)+5 for all i=3,4…n-1
f(u1v1)=1, f (u2v2)=4
f(uivi)=f(vi-2 vi-2)+5 for all i=3,4….n
Hence f is a harmonic mean labeling
Figure : 7
Case(ii) If the Quadrilateral starts from u1 then define a function f: VA(Qn) →{1,2…q+1}
by f(u1)=1, f(u1)=1, f(u2)=4
f(ui) = f(ui-2)+1 for all i=3,4…n
f(vi) = f(ui)+1 for all i=1,3…n-1
f(vi) = f(ui)-1 for all i=2,4…. N
Edges are labeled with
f(u1u2) =2, f(u2u3)=5
f(uiui+1) = f(ui-2 vi-1)+5 for all i=3,4…n-1 and f(v1v2)=3
f(vivi+1) = f (vi-2 vi-1)+5 for all i=3,4…n-1
f(u1v1) =1, f(u2v2)=4
f(uivi) = f(ui-2 vi-2)+5 for all i=3,4…n
Hence f is a Harmonic mean labeling
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Figure : 8
From case (i) and case(ii), we conclude that A(Qn) is a Harmonic mean graph.
References
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Combinatorics, 17, # DS6.
2. Harary. F. (1988), Graph theory, Narosa publishing House Reading, NewDelhi.
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Received: March, 2012