Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 4, 197 - 208
Some Results on Harmonic Mean Graphs
S. S. Sandhya1, S. Somasundaram2 and R. Ponraj3
1. Department of Mathematics
Sree Ayyappa College for women,
Chunkankadai, Kanyakumari, India Pin: 629 807
2. Department of Mathematics
Manonmaniam Sundaranar University
Tirunelveli, India Pin: 627 412
3. Department of Mathematics
Sri Paramakalyani College
Alwarkurichi, India, Pin: 627 012
sssandhya2009@gmail.com
somumsu@rediffmail.com
ponrajmath@gmail.com
Abstract
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 labels f(x) from 1,2…q+1
in such a way that when each edge e=uv is labeled with
⎡ 2 f (u ) f (v) ⎤
⎢ 2 f (u ) f (v) ⎥
f(uv) = ⎢
or ⎢
⎥
⎥ then the edge labels are distinct. In this case
⎢ f (u ) + f (v) ⎥
⎣ f (u ) + f (v) ⎦
f is called Harmonic mean labeling of G.
198
S. S. Sandhya, S. Somasundaram and R. Ponraj
In this paper we prove that Cm ∪Pn, m≥3, n>1, Cm∪Cn, m≥3, n≥3, nK3,
nK3∪Pm m >1, mC4, mC4∪Pn nK3∪mC4 and crown are harmonic mean graphs. Also
we prove that wheel Wn, Kn- e, n>4, and Prism are not harmonic mean graphs.
Mathematics Subject Classification: 05C69
Keywords: Graph, Harmonic mean graph, Crown, Wheel, Prism.
1. Introduction
All graphs in this paper are finite, simple and undirected graphs. The symbols
V(G) and E(G) will denote the vertex set and edge set of graph G. The graph G-e is
obtained from G by deleting an edge e. The union of two graphs G1 and G2 is a graph
G1∪G2 with V(G1∪G2)=V(G1)∪V(G2)and E (G1∪G2)=E(G1) ∪ E(G2). mG denotes
the disjoint union of m copies of G. The addition of two graphs G1 and G2 is a graph
G1+G2 with V(G1+G2)=V(G1)∪V(G2) and E(G1+G2) = E(G1)∪E(G2)∪ {uv:
u∈V(G1), v∈∪ V(G2)}. The graph Wn=Cn-1+K1 is called a wheel. The Cartesian
product G1xG2 of the graph G1 and G2 has the vertex set V(G1)xV(G2). Two vertices
(u1, u2) and (v1,v2) in G1xG2 are adjacent if u1 is adjacent to v1 in G1 and u2=v2 or
u1=v1 and u2 is adjacent to v2 in G2. The product graph CmxPn called a prism. The
corona G1AG2 of two graphs G1 and G2 is defined as the graph G obtained by taking
one copy of G1 (which has p1 vertices) and p1 copies of G2 and then joining the ith
vertex of G1 to every vertices in the ith copy of G2. The graph CnAK1 is called a
crown. Terms not defined here are used in the sense of Harary [2]. S. Somasundaram
and R.Ponraj introduced mean labeling of graphs in [3] and some results on mean
labelings of graphs in [4]. Harmonic mean labeling is introduced and studied for
some standard graphs in [7]. In this paper we investigate some more graphs for
harmonic mean labeling.
Some results on harmonic mean graphs
199
Note 1.1: If G is a harmonic mean graph then 1 must be a label of one of the
vertices of G, since an edge should get label 1 [7].
Note: 1.2: If G is a k-regular graph (k>2), then G is not harmonic [7].
Note 1.3: The harmonic mean of 1 and any number n lies between 1 and 2.
These observations are useful throughout.
2. Main Results
Theorem 2.1: Cm ∪Pn is a harmonic mean graph for m≥3, and n>1
Proof: Let Cm be, the cycle u1, u2…umu1 and Pn be the path v1v2…vn.
Define a function f:V(Cm∪Pn)→{1,2…q+1}
by f(ui) = i, 1 ≤i≤m
f(vi)= m+i, 1 ≤i≤n
Then the set of labels of the edges of Cm are {1,2…m} and the set of labels of the
edges of Pn is {m+1, m+2….m+n-1}
Hence Cm∪Pn is a harmonic mean graph if m≥3 and n>1.
Example
A harmonic mean labeling of C5∪C4 is given below
1
2
2
1
7
9
3
6
5
7
8
3
8
5
4
6
4
Figure (1)
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S. S. Sandhya, S. Somasundaram and R. Ponraj
Theorem 2.2: Cm∪Cn is a harmonic mean graph for m≥3 and n≥3
Proof: Let Cn be the cycle u1u2u3…..unu1 and Pn be the path v1v2….vn.
Obviously the harmonic mean labeling f in theorem 2.1 is also a harmonic mean
labeling of Cm∪Cn.
Example
A harmonic mean labeling C6∪P4 is given below
7
1
2
7
8
2
1
3
6
8
9
3
10
6
10
9
4
5
5
4
Figure (2)
In particular we have K3∪Pn for n≥2 and K3∪Cn for n≥3 are harmonic mean graphs.
Now we have the following
Theorem 2.3: nK3 is a harmonic mean graph
Proof: Let the vertex set of nK3 be V=V1∪V2 ∪….∪Vn where
Vi={vi1, vi2, vi3}
Define a function f: V(nK3) →{1,2…q+1}
by f(vij) = 3(i-1)+j, l≤i≤n, 1≤j≤3
Clearly the harmonic mean of two consecutive integers a and a+1 ies between a and
a+1.
Some results on harmonic mean graphs
201
Consider the graph with vertex labels 3i-2 3i-1 and 3i
3i-1
3i-2
3i
3i-2
3i-1
3i
Figure (3)
For the edge joining the vertices 3i-2 and 3i-1 we may assign the label 3i-2.
Similarly for the vertices 3i-1 and 3i we may assign the edge label 3i.
Since 3i-2 <
2(3i )(3i − 2)
< 3i
3i + 3i − 2
That is 3i-2 <
2(3i )(3i − 2)
< 3i
6i − 2
We may assign the edge label 3i-1, for the edge joining the vertices 3i-2 and 3i
Since the graph nK3 has distinct edge labels, it is a harmonic mean graph.
Example
A harmonic mean labeling of the graph 5K3 is
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S. S. Sandhya, S. Somasundaram and R. Ponraj
2
1
1
5
3
2
4
3 4
5
8
6
7
6
14
7
13
13
8
11
9
10
9
10
15
14
15
Figure (4)
Next we have the following
Theorem 2.4: nK3∪Pm is a harmonic mean graph for m>1
Proof: Let the vertex set of nK3 be
V=V1∪V2∪…∪Vn Where
Vi = {vi1, vi2, vi3,}
Let Pm be the path u1 u2…um
Define a function f:V(nK3∪Pm)→{1,2…q+1}
by f(vij) = 3(i-1)+j, 1≤i≤n, 1≤j≤m
f(ui) = 3n+j, 1≤j≤m.
Hence nK3∪Pm is a harmonic mean graph.
12
11
12
Some results on harmonic mean graphs
203
Example
A harmonic mean labeling of 4K3∪P7 is given below
2
1
1
5
3
2
4
3
4
8
6
5
14
7
6
7
9
8
9
16
15
13
11
10
12
10
11
12
18
16
17
18
14
13
15
17
19
Figure (5)
The same argument as in Theorem 2.4 gives the following:
Theorem 2.5: nK3∪Cm for m≥3 is a harmonic mean graphs.
Then we have the following:
Theorem 2.6: mC4 is a harmonic mean graph
Proof: Let the vertex set of mC4 be V=V1∪V2 …∪Vm where Vi= { vi1, vi2, vi3, vi4,}
Now define a function
f: V(mC4)→{1,2,3…q+1} by f(vij)= 4(i-1)+j, 1≤i≤m 1≤j≤4
If a and a+1 are two integers, then harmonic mean lies between a and a+1
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2a (a + 1)
<a+1
2a + 1
Consider a graph with vertices 4i-3, 4i-2, 4i-1 and 4i
we have a<
4i-1
4i
4i
4i-1
4i-2
4i-2
4i-3
4i-3
Figure (6)
For the edges joining the vertices 4i-3 and 4i-2 we may assign the edge label
4i-3. Similarly for the edge joining the vertices 4i-2 and 4i-1 we may assign the edge
label 4i-1 and for the edge joining the vertices 4i-1 and 4i we may assign the edge
label 4i
24i (4i − 3)
<4i-1
8i − 3
We may assign the edge label 4i-2 for the edge joining the vertices 4i-3 and 4i
Since 4i-3 <
Since mC4 has distinct edge labels, it is a harmonic mean graph.
Example
The following figure shows the harmonic mean labeling of 5C4
3
4
4
3
2
7
8
2 7
1
1
6
8
6
5
5
11
12
11
10
12
15
16
10 15
9
9
14
16
14
13
13
Some results on harmonic mean graphs
19
205
20
19
20
18
18 17
17
Figure (7)
Further we have the following
Theorem 2.7: mC4∪Pn is a harmonic mean graph
Proof: Let the vertex set of mC4 be V=V1∪V2∪…∪Vm where Vi= {vi1, vi2, vi3, vi4}
Let Pn be the path u1u2…un
Define a function f: V(mC4∪Pn)→{1,2…q+1}
by f(vij) = 4(i-1)+j, 1≤i≤m 1≤j≤4
and f(ui) = 4m+i, 1≤i≤n
Hence mC4∪Pn is a harmonic mean graph
The same argument as in Theorem 2.7 gives the following
Theorem: 2.8: mC4∪Cn n≥3 is a harmonic mean graph.
Next we have
Theorem 2.9: nK3∪mC4 is a harmonic mean graph.
Proof: Let the vertex set of nK3 be V=V1∪V2…∪Vn where Vi= {vi1, vi2, vi3}
Let the vertex set of mC4 be U=U1∪U2∪….∪Um where Uk={uk1, uk2, uk3, uk4}
Define a function f:V(nK3∪mC4}→{1,2..q+1}
by f (uij)=3(i-1)+j, 1≤i≤n, 1≤j≤3
f((ukl)=4(k-1)+3n+l, 1≤k≤m, 1≤l≤4
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S. S. Sandhya, S. Somasundaram and R. Ponraj
Hence nK3∪mC4 is a harmonic mean graph.
Example
The harmonic mean labeling of 3K3∪2C4 is given below
2
1
1
2
11
12
10
10
5
3
4
3
4
5
6
15
16
16
12
13
11
8
6
14
13
7
7
9
8
9
17
14
15
17
Figure (8)
Now we prove the following
Theorem 2.10: Crown CnAK1 is a harmonic mean graph for all n≥3
Proof: Let Cn be the Cycle u1u2….unu1 and vi be the pendant vertices adjacent to ui,
1≤i≤n
Define f: V(CnAK1)→ {1,2…q+1}
by f(ui)=2i and f(vi)=2i-1, 1≤i ≤n
Obviously f is a harmonic mean labeling
Example
Harmonic mean labeling of C6AP1 is given below
Some results on harmonic mean graphs
207
1
3
1
3
2
2
4
4
11
12
5
12
6
11
6
5
7
10
9
10
8
8
7
9
Figure (9)
Theorem 2.11: The following graphs are not harmonic mean graphs.
1. Wheels are not harmonic mean graphs.
2. CmxPn is not a harmonic mean graph.
3. Kn-e, n>4 is not a harmonic mean graph.
Proof follows from note 1.1 and 1.2
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References
[1]
J.A. Gallian, 2010, A dynamic survey of graph labeling. The Electronic
Journal of Combinatorics 17 #DS6.
[2]
Harary F, 1988, Graph Theory, Narosa publishing House Reading, New
Delhi.
[3]
S.Somasundaram and R.Ponraj, Mean Labelings of graphs, National
Academy science Letters vol: 26, (2003), 210-213.
[4]
S.Somasundaram and R. Ponraj, Some Results on Mean graphs, Pure and
applied Mathematika sciences vol: 58, (2003), 29-35
[5]
S. Somasundaram and R.Ponraj, On mean graphs of order ≤5, Journal of
Decision and mathematical sciences vol.9, (2004), 1-3
[6]
S. Somasundaram and R. Ponraj, Non-existence of mean labeling, for a
wheel, Bulletin of pure and Applied sciences vol.22E (No.1), (2003),
103-111.
[7]
S. Somasundaram and R. Ponraj ,and S.S Sandhya, Harmonic mean labelings
of graphs, communicated.
[8]
S.Somasundaram and S.S.Sandhya, Skolem Harmonic mean labelings of
graphs, to appear in Bulletin of pure and Applied sciences.
Received: August, 2011