Indian Journal of Science and Technology, Vol 8(36), DOI: 10.17485/ijst/2015/v8i36/79389, December 2015
ISSN (Print) : 0974-6846
ISSN (Online) : 0974-5645
Mean Labeling in Extended Duplicate Graph of Twig
B. Selvam* and K. Thirusangu
Department of Mathematics, S.I.V.E.T. College, Gowrivakkam, Chennai – 600 073, India; [email protected],
[email protected]
Abstract
In this paper, we prove that the extended duplicate graph of twig is mean labeling.
Keywords: Extended Duplicate Graphs, Graph Labeling, Mean Labeling
1. Introduction
Graph theory is the fast growing area of combinatorics.
Graph labeling is an assignment of integers to the vertices or edges or both subject to certain conditions. Labeled
graphs serve as useful models for a wide range of applications such as data security, communications networks,
X-ray, radar, circuit design and data base management.
After the introduction of graph labeling, various labeling
of graphs such as graceful labeling, cordial labeling, prime
cordial labeling. magic labeling, anti magic labeling etc.,
have been studied in over 1400 papers1. Graphs that are
considered in this paper are finite, simple and undirected
graph G = (V, E) having p vertices and q edges. Several
researchers refer to Rosa’s2 work.
In6,7, Somasundaram and Ponraj introduced the idea
of mean labeling of graphs. In their work, they have
shown that the graphs Pn, Cn, K2,n and K2 + mK1 are mean
graphs.
We will provide some definitions which are necessary
for this paper.
1.1 Twig
A graph G(V, E) obtained from a path by attaching
exactly two pendent edges to each internal vertices of the
path is called a Twig graph, where m is the internal vertices. In general, a twig Tm has 3m + 2 vertices and 3m + 1
edges.
*Author for correspondence
1.2 Duplicate Graph
Let G (V, E) be a simple graph and the duplicate graph of
G is DG = (V1, E1),where the vertex set V1 = V ∪ V′ and
V ∩ V′ = f and f : V → V′ is bijective (for v ∈ V, we write
f(v) = v′ for convenience) and the edge set E1 of DG is
defined as the edge ab is in E if and only if both ab′ and
a′b are edges in E1. It is evident that the duplicate graph of
path graph is disconnected.
1.3 Extended Duplicate Graph
If G (V, E) is a Twig graph Tm , the duplicate graph of G is
DG = (V1, E1) where the vertex set V1 = V ∪ V′ and V ∩ V′
= f be a duplicate graph of a Twig Tm. By adding an edge
between any one vertex from V to any other vertex in V′,
except the terminal vertices of V and V′. For convenience,
let us take v2 ∈ V and v′2 ∈ V′ and thus the edge (v2, v′2) is
formed. Call this new graph as the Extended Duplicate
Graph of the Twig Tm and it is denoted as EDG (Tm).3-5
1.4 Mean Labeling
For a graph G having p vertices and q edges is called a
mean labeling, if there is an injective function f from
the vertices of G to {0,1,2, …,q} such that each edge uv
( f (u) + f (v ))
is labeled as
if f (u) + f (v) is even and uv
2
( f (u) + f (v ) + 1)
labeled as
if f (u) + f (v) is odd, then the
2
obtaining edge labels are different.
Mean Labeling in Extended Duplicate Graph of Twig
2. Main Results
vertices v1, v2, v3, ……, v3m, v3m+1 , v3m+2 receive labeled 0,
6m + 2, 2, 4, 6, 6m, 6m – 2, 6m – 4, 8, 10, 12,……., 3m + 6,
3m + 4, 3m + 2 and the vertices v′1, v′2, v′3, v′4, v′5, v′6, v′7, v′8, …
v′3m, v′3m+1, v′3m+2 receive consecutive odd numbers labeled
6m+3, 1, 6m + 1, 6m – 1 ,6m – 3, 3, 5, 7, …….., 3m – 3,
3m – 1 and 3m + 1.
2.1 Mean Labeling
In this section, we now present an algorithm and prove
the existence of Mean labeling for EDG(Tm).
Algorithm
Procedure (mean labeling for EDG(Tm))
Case 2: If m = 2n – 1; n Œ N, the vertices v1, v2, v′ 1 and v′2
receive label 0, 6m + 2, 6m + 3 and 1 respectively; the vertices v3+j+6i receive label 6i + 2 + 2j and the vertices v′3+j+6i
V ← {v1, v2, …, v3m, v3m+1, v3m+2, v′1, v′2, … v′3m, v′3m+1, v′3m+2}
E ← {e1, e2, …, e3m, e3m+1, e3m+2, e′1, e′2, … e′3m, e′3m+1}
v1 ← 0; v2 ← 6m + 2; v′1 ← 6m + 3; v′2 ← 1;
if (m = 2n)
for i = 0 to (m – 2)/2 do
for j = 0 to 2 do
v3+j+6i ← 6i + 2 + 2j;
v6+j+6i ← 6m – 6i – 2j;
v′3+j+6i ← 6m + 6i + 1 – 2j;
v′6+j+6i ← 6i + 3 + 2j;
end for
end for
else
for i = 0 to (m – 1)/2 do
for j = 0 to 2 do
v3+j+6i ← 6i + 2 + 2j;
v′3+j+6i ← 6m + 6i + 1 – 2j;
end for
end for
for i = 0 to (m – 3)/2 do
for j = 0 to 2 do
v6+j+6i ← 6m – 6i – 2j;
v′6+j+6i ← 6i + 3 + 2j;
end for
end for
end if
end procedure
receive label 6m + 6i + 1 – 2j for 0 ≤ i ≤ (m – 1)/2 and 0 ≤
j ≤ 2; the vertices v6+j+6i receive label 6m – 6i – 2j and the
vertices v′6+j+6i receive label 6i + 3 + 2j for 0 ≤ i ≤ (m – 3)/2
and 0 ≤ j ≤ 2. Hence all the 6m + 4 vertices are labeled
such that the vertices v1, v2, v3, ,……, v3m, v3m+1 receive 0;
6m + 2, 2, 4, 6, 6m, 6m – 2, 6m – 4, 8, 10, 12, ……..,3m –
1, 3m + 1, 3m + 3 and the vertices v′1, v′2, v′3, v′4, v′5, v′6, v′7, v′8,
… v′3m, v′3m+1, v′3m+2 receive labeled 6m + 3, 1, 6m + 1, 6m –
1, 6m – 3, 3, 5, 7,…….., 3m + 4, 3m + 2, 3m.
Now we define the induced function f * : E → N as
follows
f*
0 v1  1
14 v2 
2 v3 
Case 1: If m = 2n; n Œ N, the vertices v1, v2, v′1 and v′2
receive label 0, 6m + 2, 6m + 3 and 1 respectively; the
vertices v3+j+6i receive label 6i + 2 + 2j for 0 ≤ i ≤ (m – 2)/2
and 0 ≤ j ≤ 2; the vertices v6+j+6i receive label 6m – 6i – 2j;
the vertices v′3+j+6i receive label 6m + 6i + 1 – 2j; the vertices v′6+j+6i receive label 6i + 3 + 2j. Hence all the 6m + 4
Vol 8 (36) | December 2015 | www.indjst.org
( )
if f (v i ) + f (v j ) is even
if f (v i ) + f (v j ) is odd; v i , v j ∈V.
The induced function gives the consecutive numbers
0, 1, 2 …, 3m – 1, 3m, 3m + 1, 3m + 2 for the edges e1, e2,
2
3
4 v4 
4
6 v5 
Theorem : For a twig Tm , m ≥ 1 , the extended duplicate11
12 v6 
graph of is mean labeling.
10
10 vver7 
Proof: Let Tm, m ≥ 1 be a twig. In order to label the
9
tices, define a function f : V → {1, 2, …, 6m + 3} as given
8 v8 
in algorithm.
2
 f (v i ) + f (v j )

2
vi v j = 
+ f (v j ) + 1)
(
f
(
v
)
i


2
15 v1 15
8 0 v v12
1
1
14
14 v2 v3 13 8
13
2
2 12
v3 v 4
11
3

4 v4  v 
45 9
5

6 v5 
6 v 6 3
11
12 v6 
7 v
7
10
5
10 v7 
v98
7
8 v8 
21  v1
0 v1 1
15 v1 15
20 v2
v 2
2 v3
14
13
v 
3

12 v 4

1
13
11
11
2
3
4 v4 4
6 v5
18 v6
17
21

0 v1v 2 11
20

20
19v2v3 19
 2
18
2 v3v 4 17
3



54 v4v5 415


66 v5v 6 3
16
v5
9
5
 17
16 v7

18
7 v6 
15
v7
6 v 6 3
5

 16
14
v
8


16
v
7
14 v8 15
7 v7
8
 7
5
8 v9
v 

14
v
8
13
9
8
7
 v9 813
8
10v10 
7
8 v9
10
12
1(a)
 v10 911
12v11 

(a)
(b) 10v10 10
1(a)
v11
1(b) 12v11  9
Figure 1. Mean labeling for EDG(T2) and EDG(T3).
Figure 1. Mean labeling for EDG(T2) and EDG(T3).
1(b)
Indian Journal of Science and Technology
21  v1
11
20
19
 v
2
 v
3
18  v 4

2
1
1
1
v5
1
v 6
3
v 7
5
14 v8
7
5
6
7



13
12
Figure 1. Mean labeling for EDG(T2) and EDG(T3).

 v9 1
 v10 1
v11
9
B. Selvam and K. Thirusangu
e3, …, e3m-1, e3m, e3m+1, e3m+2 and the consecutive numbers
3m + 3, 3m + 4, 3m + 5, …, 6m, 6m + 1, 6m + 2, 6m + 3 for
the edges e′3m+1, e′3m, e′3m–1, …, e′4, e′3, e′2 and e′1. Hence all the
6m + 3 edges are 1, 2, 3 …, 6m, 6m + 1, 6m + 2, 6m + 3,
which are all different.
Thus , the extended duplicate graph of Tm, m ≥ 1, is
mean labeling.
Example: Mean labeling for graphs EDG(T2) and EDG(T3)
are shown in Figures 1(a) and 1(b) respectively.
3. References
1. Gallian JA. A Dynamic Survey of graph labeling. The
Electronic Journal of combinatories. 2012; 19: # DS6.
2. Rosa A. On certain Valuations of the vertices of a graph.
Theory of graphs (Internat. Symposium, Rome, July
Vol 8 (36) | December 2015 | www.indjst.org
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5.
6.
7.
1966), Gordon and Breech, N.Y. and Dunod paris. 1967.
p. 349– 55.
Selvam B, Thirusangu K, Ulaganathan PP. Odd and Even
graceful labelings in extended duplicate twig graphs.
International Journal of Applied Engineering Research.
2011 Apr; 6(8):1075–85.
Selvam B, Thirusangu K, Ulaganathan PP. (a,d)- antimagic
labelings in extended duplicate twig graphs. Indian Journal
of Science and Technology. 2011; 4(2):112–15.
Selvam B, Thirusangu K, Ulaganathan PP. Felicitious labeling in extended duplicate twig graphs. Indian Journal of
Science and Technology. 2011; 4(5):586–89.
Somasundaram S, Ponraj R. Mean labelings of graphs.
National Academic . Science. Let. 2003; 26(7-8):210–13.
Somasundaram S, Ponraj R. Some results on mean
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Indian Journal of Science and Technology
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