Kragujevac Journal of Mathematics
Volume 37(1) (2013), Pages 163–186.
F -GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS
AND THORN GRAPHS
A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRAN
Abstract. A function f is called a F -Geometric mean labeling of a graph G(V, E)
if f : V (G) → {1, 2, 3, . . . , q + 1} is injective
and kthe induced function f ∗ : E(G) →
jp
∗
{1, 2, 3, . . . , q} defined as f (uv) =
f (u)f (v) , for all uv ∈ E(G), is bijective.
A graph that admits a F -Geometric mean labelling is called a F -Geometric mean
graph. In this paper, we have discussed the F -Geometric mean labeling of some
chain graphs and thorn graphs.
1. Introduction
Throughout this paper, by a graph we mean a finite, undirected and simple graph.
Let G(V, E) be a graph with p vertices and q edges. For notations and terminology,
we follow [3]. For a detailed survey on graph labeling, we refer [2].
Path on n vertices is denoted by Pn and a cycle on n vertices is denoted by Cn .
G ⊙ K1 is the graph obtained from G by attaching a new pendant vertex to each
vertex of G. A star graph Sm is the complete bipartite graph K1,m . G ⊙ S2 is the
graph obtained from G by attaching two pendant vertices at each vertex of G. If
(i) (i) (i)
(i)
v1 , v2 , v3 , . . . , vm+1 and u1 , u2, u3 . . . , un be the vertex of the star graph Sm and the
path Pn , then the graph (Pn ; Sm ) is obtained from n copies of Sm and the path Pn by
(i)
joining ui with the central vertex v1 of the ith copy of Sm by means of an edge, for
1 ≤ i ≤ n. The H− graph is obtained from two paths u1 , u2, . . . , un and v1 , v2 , . . . , vn
of equal length by joining an edge u n+1 v n+1 when n is odd and u n+2 v n2 when n is
2
2
2
even. Let G1 and G2 be any two graphs with p1 and p2 vertices, respectively. Then
the cartesian product G1 × G2 has p1 p2 vertices which are {(u, v)/u ∈ G1 , v ∈ G2 }.
The edges are defined as follows: (u1 , v1 ) and (u2 , v2 ) are adjacent in G1 × G2 if either
Key words and phrases. Labeling, F -Geometric mean labeling, F -Geometric mean graph.
2010 Mathematics Subject Classification. 05C78.
Received: February 18, 2012.
Revised: January 29, 2013.
163
164
A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRAN
u1 = u2 and v1 and v2 are adjacent in G2 or u1 and u2 are adjacent in G1 and v1 = v2 .
The Ladder graph Ln is obtained from Pn × P2 .
The study of graceful graphs and graceful labeling methods first introduced by Rosa
[5]. The concept of mean labeling was first introduced by S. Somasundaram and R.
Ponraj [6] and it was developed in [4] and [7]. In [10], R. Vasuki et al. discussed
the mean labeling of cyclic snake and armed crown. In [8], S. Somasundaram et al.
defined the geometric mean labeling as follows.
A graph G = (V, E) with p vertices and q edges is said to be a geometric mean graph
if it is possible to label the vertices x ∈ V with distinct labels f (x) from
jp1, 2, . . . , qk+ 1
f (u)f (v) or
in such way that when each edge e = uv is labeled with f (uv) =
lp
k
f (u)f (v) then the edge labels are distinct.
In the above definition, the readers will get some confusion in finding the edge
labels which edge is assigned by flooring function and which edge is assigned by
ceiling function.
In [9], they have given the geometric mean labeling of the graph C5 ∪ C7 as in the
Figure 1.
1q
1
6q
2
2 q
q5
3
5
q
4
3
7q
u 7
6
8
12 q
12
q
q
q
10
11 11 10
4
v q8
9
wq
9
Figure 1. A Geometric mean labeling of C5 ∪ C7 .
k
jp
f (u)f (v)
From the above figure, for the edge uv, they have used flooring function
m
lp
f (u)f (v) for fulfilling their
and for the edge vw, they have used ceiling function
requirement. To avoid the confusion of jassigning the
k edge labels in their definition,
p
we just consider the flooring function
f (u)f (v) for our discussion. Based on
our definition, the F -Geometric mean labeling of the same graph C5 ∪ C7 is given in
Figure 2.
1
1q
6
2
6q
8q
3q
q7
q9
8
9
5 10 q
3
4
q
q
4
q11
10
5
7
q 11
13
q
12
12
Figure 2. A F -Geometric mean labeling of C5 ∪ C7
In [1], A. Durai Baskar et al. introduced Geometric mean graph.
A function f is called a F -Geometric mean labeling of a graph G(V, E) if
f : V (G) → {1, 2, 3, . . . , q + 1} is injective and the induced function f ∗ : E(G) →
F -GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 165
{1, 2, 3, . . . , q} defined as
f ∗ (uv) =
k
jp
f (u)f (v) , for all uv ∈ E(G),
is bijective. A graph that admits a F -Geometric mean labeling is called a F -Geometric
mean graph.
The graph shown in Figure 3 is a F -Geometric mean graph.
17r
15
14 r
r1
1
r2
16
14
12
r
12
4
11
5
r
16
13
9r 8
9
r
r
11 10 10 7
8r
6
r
5 3
2
r3
Figure 3. A F -Geometric mean graph
In this paper, we have discussed the F -Geometric mean labeling of some chain
graphs and thorn graphs.
2. Main Results
The graph G∗ (p1 , p2 , . . . , pn ) is obtained from n cycles of length p1 , p2 , . . . , pn by
identifying consecutive cycles at a vertex as follows. If the j th cycle is of odd length,
th
p +3
vertex is identified with the first vertex of (j + 1)th cycle and if the
then its j2
th
p +2
vertex is identified with the first vertex
j th cycle is of even length, then its j2
of (j + 1)th cycle.
Theorem 2.1. G∗ (p1 , p2 , . . . , pn ) is a F -Geometric mean graph for any pj , for
1 ≤ j ≤ n.
(j)
Proof. Let {vi ; 1 ≤ j ≤ n, 1 ≤ i ≤ pj } be the vertices of the n number of cycles.
(j)
For 1 ≤ j ≤ n − 1, the j th and (j + 1)th cycles are identified by a vertex v pj +3 and
2
(j+1)
v1
(j)
while pj is odd and v pj +2 and
(j+1)
v1
2
We define f : V [G∗ (p1 , p2 , . . . , pn ] →
(1)
f (vi )
=
(
while pj is even.
1, 2, 3, . . . ,
2i − 1
2p1 − 2(i − 2)
n
P
j=1
pj + 1
)
i ≤ p21 + 1
1 p≤
1
+ 2 ≤ i ≤ p1
2
as follows:
and
166
A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRAN
for 2 ≤ j ≤ n,
(j)
f (vi )
=

j−1
P


pk + 2i − 1



k=1




j−1

P


pk + 2(i − 1)


2≤i≤
i=
k=1
j−1

P


pk + 2(i − 2)



k=1




j−1

P


pk + 2pj − 2(i − 2)

i=
i=
k=1
The induced edge labeling is as follows:

2i − 1



2i − 1
(1) (1)
f ∗ (vi vi+1 ) =
2p − 2(i − 1)


 1
2p1 − 2(i − 1)
(1)
f
∗
(j) (j)
(vi vi+1 )
=

j−1
P


pk + 2i − 1



k=1




j−1

P


pk + 2i − 1


pj 2
pj i=
k=1
j−1

P


pk + 2pj − 2(i − 1)



k=1




j−1

P


pk + 2pj − 2(i − 1)

f
∗
(j)
(vp(j)
v1 )
j
+ 2 and pj is odd
+ 2 and pj is even
+ 3 ≤ i ≤ pj
1≤i≤
k=1
and
2
=
+1
1 ≤ i ≤ p21
i = p21 + 1 and p1 is odd
i = p21 + 1 and p1 is even
p1
+ 2 ≤ i ≤ p1 − 1,
2
f ∗ (vp(1)
v1 ) = 2,
1
for 2 ≤ j ≤ n,
2
pj 2
pj j−1
X
i=
pj 2
pj pj 2
2
pj 2
+ 1 and pj is odd
+ 1 and pj is even
+ 2 ≤ i ≤ pj − 1
pk + 2.
k=1
Hence, f is a F -Geometric mean labeling of G∗ (p1 , p2 , . . . , pn ). Thus the graph
G∗ (p1 , p2 , . . . , pn ) is a F -Geometric mean graph.
A F -Geometric mean labeling of G∗ (10, 9, 12, 4, 5) is as shown in Figure 4.
F -GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 167
5q
5
3q 3
1 q 1
2
4 q
4
q
6
q7
8
6
q
8
17
q
26
q
q
24 26 28
28
q30 32
22
22q
19
30
20
q20
q 32
21
31
q
10 12
18 q
29 31 33
q 23 23q
q
q
q
16
14
18
25
27
10 14
q
q
25
29
27
16
7 q 9 15q 15
12q13
9
11
q11
17
q19
24q
34
q
40
q
34
38q 38
40
36
q36
35
q 41
37
q
39
q
39
35
Figure 4. A F -Geometric mean labeling of G (10, 9, 12, 4, 5)
∗
The graph G′ (p1 , p2 , . . . , pn ) is obtained from n cycles of length p1 , p2 , . . . , pn by iden
th
p +3
tifying consecutive cycles at an edge as follows: The j2
edge of j th cycle is
th
pj +1
th
edge
identified with the first edge of (j + 1) cycle when j is odd and the
2
of j th cycle is identified with the first edge of (j + 1)th cycle when j is even.
Theorem 2.2. G′ (p1 , p2 , . . . , pn ) is a F -Geometric mean graph if all pj ’s are odd or
all pj ’s are even, for 1 ≤ j ≤ n.
(j)
Proof. Let {vi ; 1 ≤ j ≤ n, 1 ≤ i ≤ pj } be the vertices of the n number of cycles.
Case (i) pj is odd, for 1 ≤ j ≤ n.
For 1 ≤ j ≤ n − 1, the j th and (j + 1)th cycles are identified by the edges
(j)
(j)
(j+1) (j+1)
(j)
(j)
(j+1) (j+1)
v pj+1 v pj+3 and v1
vpj+1 while j is odd and v pj−1 v pj+1 and v1
vpj+1 while j is
2
2
2
2
even.
(
)
n
P
We define f : V [G′ (p1 , p2 , . . . , pn )] → 1, 2, 3, . . . ,
pj − n + 2 as follows:
j=1
and for 2 ≤ j ≤ n,
(j)
f (vi )
=

 1
(1)
2i
f (vi ) =

2p1 + 3 − 2i

j−1
P


pk − j + 2i + 2



k=1




j−1

P


pk + 2pj + 3 − j − 2i


k=1
j−1

P


pk − j + 2i + 1



k=1




j−1

P


pk + 2pj + 4 − j − 2i

k=1
i=1 2 ≤i ≤ p21 + 1
p1
+ 2 ≤ i ≤ p1
2
2≤i≤
pj 2
2
2
and j is even
+ 1 ≤ i ≤ pj − 1 and j even
2≤i≤
pj pj pj 2
+ 1 and j is odd
+ 2 ≤ i ≤ pj − 1 and j odd.
168
A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRAN
The induced edge labeling is as follows:
2i
∗ (1) (1)
f (vi vi+1 ) =
2p1 + 1 − 2i
(1)
f ∗ (vp(1)
v1 ) = 1
1
and for 2 ≤ j ≤ n,















∗ (j) (j)
f (vi vi+1 ) =














j−1
P
k=1
j−1
P
k=1
j−1
P
k=1
j−1
P
k=1
1 ≤i ≤ p21
p1
+ 1 ≤ i ≤ p1 − 1,
2
pk − j + 2i + 2
1≤i≤
pk + 2pj + 1 − j − 2i
pj pk − j + 2i + 1
1≤i≤
pk + 2pj + 2 − j − 2i
pj 2
2
pj 2
and j is even
+ 1 ≤ i ≤ pj − 1 and j even
pj 2
and j is odd
+ 1 ≤ i ≤ pj − 1 and j odd.
Case (ii) pj is even, for 1 ≤ j ≤ n.
For 1 ≤ j ≤ n − 1, the j th and (j + 1)th cycles are identified by the edges
(j+1) (j+1)
(j) (j)
vpj+1 .
v pj v pj +2 and v1
2
2
(
)
n
P
We define f : V [G′ (p1 , p2 , . . . , pn )] → 1, 2, 3, . . . ,
pj − n + 2 as follows:
j=1
and for 2 ≤ j ≤ n,
(j)

 1
(1)
2i
f (vi ) =

2p1 + 3 − 2i
f (vi ) =
 j−1
P


pk − j + 2i + 1


k=1
j−1

P


pk + 2pj + 4 − j − 2i

k=1
The induced edge labeling is as follows:
2i
∗ (1) (1)
f (vi vi+1 ) =
2p1 + 1 − 2i
(1)
f ∗ (vp(1)
v1 ) = 1
1
i=1 i ≤ p21
2 p≤
1
+ 1 ≤ i ≤ p1
2
2≤i≤
pj 2
pj 2
+ 1 ≤ i ≤ pj − 1.
1 ≤i ≤ p21
p1
+ 1 ≤ i ≤ p1 − 1,
2
F -GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 169
and for 2 ≤ j ≤ n,
(j) (j)
f ∗ (vi vi+1 ) =
 j−1
P


pj − j + 2i + 1


1≤i≤
k=1
j−1

P


pk + 2pj + 2 − j − 2i

pj 2
k=1
pj 2
+ 1 ≤ i ≤ pj − 1.
Hence, f is a F -Geometric mean labeling of G′ (p1 , p2 , . . . , pn ). Thus the graph
G′ (p1 , p2 , . . . , pn ) is a F -Geometric mean graph.
A F -Geometric mean labeling of G′ (7, 5, 9, 13) and G′ (4, 8, 10, 6) is as shown in
Figure 5.
6q
4q
4
13
q8
7
q7
q
1
3
q
3
q11
11 q
12
12
8
5
q
5
10
15 15
23
q
16
q
7q
7
5
4
q
1
3
q
10
q
2
1
27q
25q
25
27 q29
16 q18 23q 23
29
18 21
q20
q31
19
q19
31
q
20
30
q
q 32
17
28
22
22 q 24
q 30
q
q 26 28
24
17
26
14 q 14
6
2
1
16
q
9
14q
12
9
q4
q5
14
16
q18
8
q
20
19
q
10 15
q
8
q25
q20
q21
11 q11
12
10 13
6
23
21
18
q
q
3
q9
15
q
17
′
17
q
19
25
22
q
24
q
26
24
Figure 5. F -Geometric mean labeling of G (7, 5, 9, 13) and G′ (4, 8, 10, 6)
b 1 , m1 , p2 , m2 , . . . , mn−1 , pn ) is obtained from n cycles of length
The graph G(p
p1 , p2 , . . . , pn and (n − 1) paths on m1 , m2 , . . . , mn−1 vertices respectively by identifying a cycle and a path at a vertex alternatively as follows: If the j th cycles is of
th
p +3
odd length, then its j2
vertex is identified with a pendant vertex of j th path
th
pj +2
th
vertex is identified with a
and if the j cycle is of even length, then its
2
pendant vertex of j th path while the other pendant vertex of the j th path is identified
with the first vertex of the (j + 1)th cycle.
b 1 , m1 , p2 , m2 , . . . , mn−1 pn ) is a F -Geometric mean graph for any
Theorem 2.3. G(p
pj ’s and mj ’s.
(j)
(j)
Proof. Let {vi ; 1 ≤ j ≤ n, 1 ≤ i ≤ pj } and {ui ; 1 ≤ j ≤ n − 1, 1 ≤ i ≤ mj } be the
n number of cycles and (n − 1) number of paths respectively.
170
A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRAN
(j)
For 1 ≤ j ≤ n − 1, the j th cycle and j th path are identified by a vertex v pj +2 and
2
(j)
u1
(j)
while pj is even and v pj +3 and
(j)
u1
while pj is odd. And the j
th
path and (j + 1)th
2
(j)
(j+1)
cycle are identified by a vertex umj and v1
.
n
n−1
b 1 , m1 , p2 , m2 , . . . , mn−1 , pn )] → 1, 2, 3, . . . , P (pj + mj ) + pn −
We define f : V [G(p
j=1
o
n + 2 as follows:
2i − 1
1 ≤i ≤ p21 + 1
(1)
f (vi ) =
p1
2p1 + 4 − 2i
+ 2 ≤ i ≤ p1 ,
2
(1)
f (ui ) = p1 + i, for 2 ≤ i ≤ m1 ,
for 2 ≤ j ≤ n,

j−1
P



(pk + mk ) + 2i − j



k=1



j−1

P



(pk + mk ) + 2i − j − 1



k=1


(j)
f (vi ) =
2≤i≤
i=
j−1

P


(pk + mk ) + 2i − j − 3



k=1








j−1

P


(pk + mk ) + 2pj − 2i − j + 5

(j−1)
f (ui
)
=
j−2
X
k=1
f ∗ (vp(1)
v1 ) = 2,
1
(1) (1)
+1
+2
i=
pj 2
+ 2 and
pj is even
pj 2
+ 3 ≤ i ≤ pj
(pk + mk ) + pj−1 + i + 2 − j, for 2 ≤ i ≤ mj−1
The induced edge labeling is as follows:

2i − 1




2i − 1



(1) (1)
f ∗ (vi vi+1 ) =
2p1 − 2i + 2







2p1 − 2i + 2
(1)
2
2
and pj is odd
k=1
and for 3 ≤ j ≤ n,
pj pj 1 ≤ i ≤ p21
i = p21 + 1 and
p1 is odd
i = p21 + 1 and
p1p1is even
+ 2 ≤ i ≤ p1 − 1,
2
f ∗ (ui ui+1 ) = p1 + i, for 1 ≤ i ≤ m1 − 1,
F -GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 171
for 2 ≤ j ≤ n,

j−1
P



(pk + mk ) + 2i − j



k=1



j−1

P



(pk + mk ) + 2i − j


k=1



(j) (j)
f ∗ (vi vi+1 ) =
f
(j)
(vp(j)
v1 )
j
i=
j−1
X
=
k=1
pj pj 2
2
+ 1 and
pj is odd
j−1

P


(pk + mk ) + 2pj − 2i − j + 3



k=1








j−1

P


(pk + mk ) + 2pj − 2i − j + 3

k=1
∗
1≤i≤
i=
pj 2
+ 1 and
pj is even
pj 2
+ 2 ≤ i ≤ pj − 1,
(pk + mk ) − j + 3
and for 3 ≤ j ≤ n,
f
∗
(j−1) (j−1)
(ui
ui+1 )
=
j−2
X
k=1
(pk + mk) + pj−1 + i + 2 − j, for 1 ≤ i ≤ mj−1 − 1.
b 1 , m1 , p2 , m2 . . . , mn−1 , pn ). Thus the
Hence, f is a F -Geometric mean labeling of G(p
b 1 , m1 , p2 , m2 . . . , mn−1 , pn ) is a F -Geometric mean graph.
graph G(p
b 4, 5, 6, 10) is as shown in Figure 6.
A F -Geometric mean labeling of G(8,
5r
3
r
3
r
r7
9
2
4
14r 14
8
r
r
4
6
8
26r 26
16
24 r24
22
17 r17 r18 r19 r20 r21 r22
21
19
18
20
23
r
15
25 25
12
7
1
1
28
r
16
r
5
r 9 r10 r
10 11
11
r12
13
r
r
6
15
28
r30
30
r32
31
r
29 31
r27 r29
27
b 4, 5, 6, 10)
Figure 6. A F -Geometric mean labeling of G(8,
Theorem 2.4. Cn ⊙ K1 is a F -Geometric mean graph, for n ≥ 3.
Proof. Let v1 , v2 , . . . , vn be the vertices of the cycle Cn and let ui be the pendant
vertices attached
vi , for 1 ≤ i ≤ n. Consider the graph Cn ⊙ K1 , for n ≥ 4.
√ at each
2n + 1 is odd.
Case (i)
172
A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRAN
We define f : V [Cn ⊙ K1 ] → {1, 2, 3, . . . , 2n + 1} as follows:

1
i=1 j


k
√

2n+1
+ 1 and
2i
2≤i≤
f (vi ) =
2
j√
k


2n+1
 2i + 1
+2≤i≤n
2

2
i=1 j


k
√



2 ≤ i ≤ 2n+1
 2i − 1
j√
k2
f (ui ) =
2n+1
2i + 1
i=
+1



k2
j√


2n+1
 2i
+ 2 ≤ i ≤ n.
2
The induced edge labeling is as follows:

j√
k
2n+1
 2i
1
≤
i
≤
2
j√
k
f ∗ (vi vi+1 ) =
2n+1
 2i + 1
+ 1 ≤ i ≤ n − 1,
2
k
j√
2n + 1 and
f ∗ (v1 vn ) =

j√
k
2n+1
 2i − 1
1
≤
i
≤
2
k
j√
f ∗ (ui vi ) =
2n+1
 2i
+ 1 ≤ i ≤ n.
2
√
Case (ii)
2n + 1 is even.
We define f : V [Cn ⊙ K1 ] → {1, 2, 3, . . . , 2n + 1} as follows:

1
i=1 j


k
√

2n+1
2i
2≤i≤
and
f (vi ) =
2
k
j√


2n+1
 2i + 1
+1≤i≤i≤n
2

2
i=1 j


k
√

2n+1
2i
−
1
2
≤
i
≤
f (ui ) =
2
k
j√


2n+1
 2i
+ 1 ≤ i ≤ n.
2
The induced edge labeling is as follows:

 2i
∗
f (vi vi+1 ) =
 2i + 1
k
j√
f ∗ (v1 vn ) =
2n + 1

 2i − 1
f ∗ (ui vi ) =
 2i
j√
k
2n+1
1≤i≤
−1
2
k
j√
2n+1
≤ i ≤ n − 1,
2
and
j√
k
1 ≤ i ≤ 2n+1
2
j√
k
2n+1
+ 1 ≤ i ≤ n.
2
F -GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 173
Hence, the graph Cn ⊙ K1 , for n ≥ 4 admits F -Geometric mean labeling.
For n = 3, a F -Geometric mean labeling of C3 ⊙ K1 is as shown in Figure 7.
1r
1
r2
3
2
6
r
r
r
5
5
4 4
7
Figure 7. A F - Geometric mean labeling
A F -Geometric mean labeling of C12 ⊙ K1 is as shown
2r
1
3r
2
3
7 r
4
6
8 r 8
10
r
r
r
7
r
1
4
6
r9
9
r11
10
11 r
13
r
12
12
13
15
r
r
6
of C3 ⊙ K1
in Figure 8.
r24
5
24
25 23 22 22
r
23
21
21 r 20 r20
19
19 r
18
r
17
17r
18
16
r
15
16
r
r
14
r
14
Figure 8. A F -Geometric mean labeling of C12 ⊙ K1
Theorem 2.5. Cn ⊙ S2 is a F -Geometric mean graph for n ≥ 3.
(i)
Proof. Let u1 , u2 , . . . , un be the vertices of the cycle Cn . Let v1 be the pendant
vertices at each vertex ui , for 1 ≤ i ≤ n. Therefore,
(i)
and
(i)
V [Cn ⊙ S2 ] = V (Cn ) ∪ {v1 , v2 ; 1 ≤ i ≤ n}
(i)
(i)
E[Cn ⊙ S2 ] = E(Cn ) ∪ {ui v1 , ui v2 ; 1 ≤ i ≤ n}.
174
A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRAN
√ Case (i)
6n is a multiple of 3.
We define f : V [Cn ⊙ S2 ] → {1, 2, 3, . . . , 3n + 1} as follows:
j√ k

6n

3i
−
1
1
≤
i
≤

3



j√ k
3i + 1
i = 36n + 1
f (ui ) =


j√ k


 3i
6n
+ 2 ≤ i ≤ n,
3

j√ k

 3i − 2
1 ≤ i ≤ 36n
(i)
f (v1 ) =
j√ k

6n
 3i − 1
+1≤i≤n
3
and
(i)
f (v2 ) =

 3i
 3i + 1
The induced edge labeling is as follows

 3i
f ∗ (ui ui+1 ) =
 3i + 1
j√ k
∗
6n ,
f (un u1 ) =

 3i − 2
(i)
∗
f (ui v1 ) =
 3i − 1
and
(i)
f ∗ (ui v2 ) =

 3i − 1
 3i
j√ k
1 ≤ i ≤ 36n + 1
j√ k
6n
+ 2 ≤ i ≤ n.
3
j√ k
1 ≤ i ≤ 36n − 1
j√ k
6n
≤ i ≤ n − 1,
3
j√ k
1 ≤ i ≤ 36n
j√ k
6n
+1≤i≤n
3
j√ k
1 ≤ i ≤ 36n
j√ k
6n
+ 1 ≤ i ≤ n.
3
√ Case (ii)
6n is not a multiple of 3.
We define f : V [Cn ⊙ S2 ] → {1, 2, 3, . . . , 3n + 1} as follows:

j√ k
 3i − 1
1 ≤ i ≤ 36n
j√ k
f (ui ) =
6n
 3i
+ 1 ≤ i ≤ n,
3

k
j√
6n+1
 3i − 2
1
≤
i
≤
3
(i)
j√
k
f (v1 ) =
6n+1
 3i − 1
+1≤i≤n
3
F -GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 175
and
(i)
f (v2 ) =

 3i
j√ k
1 ≤ i ≤ 36n
j√ k
6n
+ 1 ≤ i ≤ n.
3
 3i + 1
The induced edge labeling is as follows

 3i
f ∗ (ui ui+1 ) =
 3i + 1
j√ k
f ∗ (un u1 ) =
6n ,

 3i − 2
(i)
∗
f (ui v1 ) =
 3i − 1
and
f
∗
(i)
(ui v2 )
=
j√ k
1 ≤ i ≤ 36n
j√ k
6n
+ 1 ≤ i ≤ n − 1,
3
k
j√
1 ≤ i ≤ 6n+1
3
j√
k
6n+1
+1≤i≤n
3

 3i − 1
j√ k
1 ≤ i ≤ 36n
j√ k
6n
+ 1 ≤ i ≤ n.
3
 3i
Hence, f is a F -Geometric mean labeling of Cn ⊙ S2 . Thus the graph Cn ⊙ S2 is a
F -Geometric mean graph, for n ≥ 3.
A F -Geometric mean labeling of C6 ⊙ S2 is as shown in Figure 9.
1
q
3q
2
1
19q
q
6
17 q
17
3
2
18
5
q
18
q
16
16 q 15
14
q
14
q4
4
10
q15
5
q6
7
8
q8
q
9
13
q
9
10
12q
11
12
q
q
13
11
Figure 9. A F -Geometric mean labeling of C6 ⊙ S2
Theorem 2.6. (Pn ; Sm ) is a F -Geometric mean graph, for m ≤ 2 and n ≥ 1.
(i)
(i)
(i)
Proof. Let u1 , u2, . . . , un be the vertices of the path Pn and Let v1 , v2 , . . . , vm+1
(i)
be the vertices of the star graph Sm such that v1 is the central vertex of Sm , for
1 ≤ i ≤ n.
176
A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRAN
Case (i) m = 1
We define f : V [(Pn ; Sm )] → {1, 2, 3, . . . , 3n} as follows:
3i
1 ≤ i ≤ n and i is odd
f (ui ) =
3i − 2
1 ≤ i ≤ n and i is even,
(i)
f (v1 ) = 3i − 1, f or
and
(i)
f (v2 )
=
3i − 2
3i
1≤i≤n
1 ≤ i ≤ n and i is odd
1 ≤ i ≤ n and i is even.
The induced edge labeling is as follows
and
f ∗ (ui ui+1 ) = 3i, f or
3i − 1
(i)
∗
f (ui v1 ) =
3i − 2
f
∗
(i) (i)
(v1 v2 )
=
1 ≤ i ≤ n − 1,
1 ≤ i ≤ n and i is odd
1 ≤ i ≤ n and i is even
3i − 2
3i − 1
1 ≤ i ≤ n and i is odd
1 ≤ i ≤ n and i is even.
Case (ii) m = 2
We define f : V [(Pn ; Sm )] → {1, 2, 3, . . . , 4n} as follows:
4i
1 ≤ i ≤ n and i is odd
f (ui ) =
4i − 2
1 ≤ i ≤ n and i is even,
(i)
f (v1 ) = 4i − 1, f or
1 ≤ i ≤ n,
f (v2 ) = 4i − 3, f or
1≤i≤n
(i)
and
(i)
f (v3 )
=
4i − 2
4i
1 ≤ i ≤ n and i is odd
1 ≤ i ≤ n and i is even.
The induced edge labeling is as follows:
f ∗ (ui ui+1 ) = 4i, f or
4i − 1
(i)
∗
f (ui v1 ) =
4i − 2
(i) (i)
and
f ∗ (v1 v2 ) = 4i − 3,
f
∗
(i) (i)
(v1 v3 )
=
4i − 2
4i − 1
1 ≤ i ≤ n − 1,
1 ≤ i ≤ n and i is odd
1 ≤ i ≤ n and i is even
for 1 ≤ i ≤ n
1 ≤ i ≤ n and i is odd
1 ≤ i ≤ n and i is even.
Hence, f is a F -Geometric mean labeling of (Pn ; Sm .) Thus the graph (Pn ; Sm ) is
a F -Geometric mean graph, for m ≤ 2 and n ≥ 1.
A F -Geometric mean labeling of (P7 ; S1 ) and (P8 ; S2 ) is as shown in Figure 10.
F -GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 177
3s
4s
3
2
2
5
1
2
s
16
20
s
s 20
11
13
17
19
s
s
s
1
6
7
12
16 20
18
19
5
7
s
24 28s
s
28
30s
14
19
22
27
30
s 11
s15
s 19
s23
s
27
s31
10 13
s s
13
20 22
11
7 9
5
12 14
s
12
s
8
s
s
17
s
21
s
s
s
7
14
s
18
s
s3
s
11
15 16s
14
10
s
5
6
2
s
s
3
1
8
s
12 15
s
10
s
6
4
9
8
4
s
1
4s
9s
6
9
s s
13
15 17
s s17
18 21
23 25
s s
21
s s25
26 29
s s 29
18
16
10
8
26
24
Figure 10. A F -Geometric mean labeling of (P7 ; S1 ) and (P8 ; S2 )
Theorem 2.7. For a H−graph G, G ⊙ K1 is a F -Geometric mean graph.
Proof. Let u1 , u2 , . . . , un and v1 , v2 , . . . , vn be the vertices of G. Therefore
V (G ⊙ K1 ) = V (G) ∪ {u′i , vi′ ; 1 ≤ i ≤ n}
and
E(G ⊙ K1 ) = E(G) ∪ {ui u′i , vi vi′ ; 1 ≤ i ≤ n}.
Case (i) n ≡ 0(mod 4).
We define f : V (G ⊙ K1 ) → {1, 2, 3, . . . , 4n} as follows:
f (ui ) =
2i − 1
2i
1 ≤ i ≤ n and i is odd
1 ≤ i ≤ n and i is even,
2n − 3 + 4i
1 ≤ i ≤ n2 − 1 and i is odd
f (vi ) =
2n − 1 + 4i
1 ≤ i ≤ n2 and i is even,
2n − 2 + 4i
1 ≤ i ≤ n2 − 1 and i is odd
f (vn+1−i ) =
2n + 4i
1 ≤ i ≤ n2 and i is even,
f (u′i )
=
2i
2i − 1
1 ≤ i ≤ n and i is odd
1 ≤ i ≤ n and i is even,
31
s
32
178
and
A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRAN

f (vi ) + 2



f (vi ) − 2
f (vi′ ) =
f (vi ) − 2



f (vi ) + 2
1 ≤ i ≤ n2 − 1 and i is odd
n
+ 1 ≤ i ≤ n and i is odd
2
1 ≤ i ≤ n2 and i is even
n
+ 2 ≤ i ≤ n and i is even.
2
The induced edge labeling is as follows:
f ∗ (ui ui+1 ) = 2i, f or 1 ≤ i ≤ n − 1,
f ∗ (ui u′i ) = 2i − 1, f or 1 ≤ i ≤ n,
and
jnk
f ∗ (vi vi+1 ) = 2n − 1 + 4i, f or 1 ≤ i ≤
,
2
jnk
− 1,
f ∗ (vn+1−i vn−i ) = 2n + 4i, f or 1 ≤ i ≤
2 
n
f (vi )


1 n≤ i ≤ 2 − 1 and i is odd

+ 1 ≤ i ≤ n and i is odd
f (vi ) − 2
2
f ∗ (vi vi′ ) =
f (vi ) − 2
1 ≤ i ≤ n2 and i is even



n
+ 2 ≤ i ≤ n and i is even
f (vi )
2
∗
f (ui+1 vi ) = 2n, f or i =
jnk
2
.
Case (ii) n ≡ 1(mod 4).
We define f : V (G ⊙ K1 ) → {1, 2, 3, . . . , 4n} as follows:
and
f (ui ) = 2i, f or 1 ≤ i ≤ n, f (u′i ) = 2i − 1, f or 1 ≤ i ≤ n,
2n − 3 + 4i
1 ≤ i ≤ n2 + 1 and i is odd
f (vi ) =
2n − 1 + 4i
1 ≤ i ≤ n2 and i is even,
2n − 2 + 4i
1 ≤ i ≤ n2 and i is odd
f (vn+1 − i) =
2n + 4i
1 ≤ i ≤ n2 and i is even

f (vi ) + 2



f (vi ) + 1
f (vi′ ) =
f (vi ) + 2



f (vi ) − 2
1 ≤ i ≤ n2 and i is odd
i = n2 + 1 and i is odd
n
+ 3 ≤ i ≤ n and i is odd
2
1 ≤ i ≤ n and i is even.
The induced edge labeling is as follows:
f ∗ (ui ui+1 ) = 2i, f or 1 ≤ i ≤ n − 1,
f ∗ (ui u′i ) = 2i − 1, f or 1 ≤ i ≤ n,
jnk
,
f ∗ (vi vi+1 ) = 2n − 1 + 4i, f or 1 ≤ i ≤
jnk 2
f ∗ (vn+1−i vn−i ) = 2n + 4i, f or 1 ≤ i ≤
,
2
F -GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 179
f
∗
(vi vi′ )
=
and
f (vi )
f (vi ) − 2
1 ≤ i ≤ n and i is odd
1 ≤ i ≤ n and i is even
f ∗ (ui vi ) = 2n, f or i =
jnk
2
+ 1.
Case (iii) n ≡ 2(mod 4).
We define f : V (G ⊙ K1 ) → {1, 2, 3, . . . , 4n} as follows:
2i
1 ≤ i ≤ n and i is odd
f (ui ) =
2i − 1
1 ≤ i ≤ n and i is even,
2i − 1
1 ≤ i ≤ n and i is odd
f (u′i ) =
2i
1 ≤ i ≤ n and i is even,
2n − 1 + 4i
1 ≤ i ≤ n2 and i is odd
f (vi ) =
2n − 3 + 4i
1 ≤ i ≤ n2 − 1 and i is even,
2n + 4i
1 ≤ i ≤ n2 and i is odd
f (vn+1−i ) =
2n − 2 + 4i
1 ≤ i ≤ n2 − 1 and i is even,
and

f (vi ) − 2



f
(vi ) + 2
f (vi′ ) =
f (vi ) + 2



f (vi ) − 2
1 ≤ i ≤ n2 and i is odd
n
+ 2 ≤ i ≤ n and i is odd
2
n
1 n≤ i ≤ 2 − 1 and i is even
+ 1 ≤ i ≤ n and i is even
2
The induced edge labeling is as follows:
f ∗ (ui ui+1 ) = 2i, f or 1 ≤ i ≤ n − 1,
f ∗ (ui u′i ) = 2i − 1, f or 1 ≤ i ≤ n,
and
jnk
f ∗ (vi vi+1 ) = 2n − 1 + 4i, f or 1 ≤ i ≤
,
jnk 2
f ∗ (vn+1−i vn−i ) = 2n + 4i, f or 1 ≤ i ≤
− 1,
2 
n
f (vi ) − 2


1 n≤ i ≤ 2 and i is odd

+ 2 ≤ i ≤ n and i is odd
f (vi )
2
f ∗ (vi vi′ ) =
f (vi )
1 ≤ i ≤ n2 − 1 and i is even



n
+ 1 ≤ i ≤ n and i is even
f (vi ) − 2
2
∗
f (ui+1 vi ) = 2n, f or i =
jnk
2
+ 1.
180
A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRAN
Case (iv) n ≡ 3(mod 4).
We define f : V (G ⊙ K1 ) → {1, 2, 3, . . . , 4n} as follows:
and
f (ui ) = 2i, f or 1 ≤ i ≤ n, f (u′i ) = 2i − 1, f or 1 ≤ i ≤ n,
2n − 1 + 4i
1 ≤ i ≤ n2 and i is odd
f (vi ) =
2n − 3 + 4i
1 ≤ i ≤ n2 + 1 and i is even,
2n + 4i
1 ≤ i ≤ n2 and i is odd
f (vn+1−i ) =
2n − 2 + 4i
1 ≤ i ≤ n2 and i is even,

f (vi ) − 2



f (vi ) + 1
f (vi′ ) =
f (vi ) − 2



f (vi ) + 2
1 ≤ i ≤ n2 and i is odd
i = n2 + 1 and i is odd
n
+ 3 ≤ i ≤ n and i is odd
2
1 ≤ i ≤ n and i is even.
The induced edge labeling is as follows:
f ∗ (ui ui+1 ) = 2i, f or 1 ≤ i ≤ n − 1,
f ∗ (ui u′i ) = 2i − 1, f or 1 ≤ i ≤ n,
∗
jnk
,
f (vi vi+1 ) = 2n − 1 + 4i, f or 1 ≤ i ≤
jnk 2
f ∗ (vn+1−i vn−i ) = 2n + 4i, f or 1 ≤ i ≤
,
2
f (vi ) − 2
1 ≤ i ≤ n and i is odd
∗
′
f (vi vi ) =
f (vi ) + 2
1 ≤ i ≤ n and i is even
and
f ∗ (ui vi ) = 2n, f or i =
jnk
2
+ 1.
Hence, f is a F -Geometric mean labeling of G ⊙ K1 . Thus the graph G ⊙ K1 is a
F -Geometric mean graph.
A F -Geometric mean labeling of G ⊙ K1 is as shown in Figure 11.
Theorem 2.8. For a H−graph G, G ⊙ S2 is a F -Geometric mean graph.
Proof. Let u1 , u2 , . . . , un and v1 , v2 , . . . , vn be the vertices of G. Therefore,
V [G ⊙ S2 ] = V (G) ∪ {u′i , u′′i , vi′ , vi′′ ; 1 ≤ i ≤ n}
and
E[G ⊙ S2 ] = E(G) ∪ {ui u′i , ui u′′i , vi vi′ , vi vi′′ ; 1 ≤ i ≤ n}.
F -GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 181
1 s
3
s
1
3
s
2 17 s
2 17
s4
4
5
7
9
11
13
5
19 s
21
15
19
s
15
s21
1 s
4
s
1
s
3
2 15 s
2 15
s3
17 s
19
4
5
23
17
s23
s6
s 21
5 s
25 s
23
★
6 25
6 ★23
★
7
s
s8 14 s 27
7
s 28
★
s
s 12 s 22
8
27
24
7
8 26
8 20
9
24 s
s 9
s10
18
s10
24
26 s
9 s
18 s
10 22
10 16
s 11
s12 20 s 20
s22
s 11
s11 16 s 14
12
12 18
s 13
s
14 18 s 16 s16
Figure 11. A F -Geometric mean labeling of H graph G ⊙ K1
s
s6
Case (i) n is odd.
We define f : V (G ⊙ S2 ) → {1, 2, 3, . . . , 6n} as follows:
f (ui ) = 3i − 1, f or 1 ≤ i ≤ n,
f (u′i ) = 3i − 2, f or 1 ≤ i ≤ n,
f (u′′i ) = 3i, f or 1 ≤ i ≤ n,
f (vi ) = 3n − 2 + 6i, f or 1 ≤ i ≤
and
13
jnk
2k
jn
,
f (vn+1−i ) = 3n − 3 + 6i, f or 1 ≤ i ≤
+ 1,

n 2
1 ≤ i ≤ 2
 f (vi ) − 2
′
n
f (vi ) =
f (vi ) − 1
i n= 2 + 1

+2≤i≤n
f (vi ) + 2
2
f (vi′′ )
=
f (vi ) + 2
f (vi ) − 2
The induced edge labeling is as follows:
1 ≤ i ≤ n2
n
+ 1 ≤ i ≤ n.
2
f ∗ (ui ui+1 ) = 3i, f or 1 ≤ i ≤ n − 1,
f ∗ (ui u′i ) = 3i − 2, f or 1 ≤ i ≤ n,
f ∗ (ui u′′i ) = 3i − 1, f or 1 ≤ i ≤ n,
jnk
∗
+ 1,
f (ui vi ) = 3n, f or i =
2
s
13
s19
s21
s 22
s 20
s14
182
A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRAN
f ∗ (vi vi+1 ) = 3n + 6i, f or 1 ≤ i ≤
and
jnk
,
2 j k
n
f ∗ (vn+1−i vn−i ) = 3n − 1 + 6i, f or 1 ≤ i ≤
,

n 2
1 ≤ i ≤ 2
 f (vi ) − 2
∗
′
f (vi vi ) =
f (vi ) − 1
i = n2 + 1

n
+2≤i≤n
f (vi )
2
f
∗
(vi vi′′ )
=
f (vi )
f (vi ) − 2
1 ≤ i ≤ n2
n
+ 1 ≤ i ≤ n.
2
Case (ii) n is even.
We define f : V (G ⊙ S2 ) → {1, 2, 3, . . . , 6n} as follows:
f (ui ) = 3i − 1, f or 1 ≤ i ≤ n,
f (u′i ) = 3i − 2, f or 1 ≤ i ≤ n,
f (u′′i ) = 3i, f or 1 ≤ i ≤ n − 1,
f (u′′n ) = 3n + 1,
f (vi ) = 3n + 1 + 6i, f or 1 ≤ i ≤
jnk
− 1,
2j k
n
f (vn+1−i ) = 3n + 6(i − 1), f or 2 ≤ i ≤
+ 1,
2
f (vn ) = 3n + 2,
f (vi ) − 2
1 ≤ i ≤ n2
′
f (vi ) =
n
f (vi ) + 2
+ 1 ≤ i ≤ n − 1,
2
f (vn′ ) = f (vn ) + 1
and

 f (vi ) + 2
′′
f (vi ) =
f (vi ) − 1

f (vi ) − 2
The induced edge labeling is as follows:
1 ≤ i ≤ n2 − 1
i = n2
n
+ 1 ≤ i ≤ n.
2
f ∗ (ui ui+1 ) = 3i, f or 1 ≤ i ≤ n − 1,
f ∗ (ui u′i ) = 3i − 2, f or 1 ≤ i ≤ n,
f ∗ (ui u′′i ) = 3i − 1, f or 1 ≤ i ≤ n,
jnk
f ∗ (ui+1 vi ) = 3n + 1, f or i =
,
2
∗
f (vi vi+1 ) = 3n + 3 + 6i, f or 1 ≤ i ≤
jnk
2
,
F -GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 183
f ∗ (vn+1−i vn−i ) = 3n − 4 + 6i, f or 2 ≤ i ≤
f ∗ (vn vn−1 ) = 3n + 3,
f (vi ) − 2
∗
′
f (vi vi ) =
f (vi )
2
,
1 ≤ i ≤ n2
n
+1≤i≤n
2
and

 f (vi )
f (vi ) − 1
f ∗ (vi vi′′ ) =

f (vi ) − 2
jnk
1 ≤ i ≤ n2 − 1
n
i n= 2
+ 1 ≤ i ≤ n.
2
Hence, f is a F -Geometric mean labeling of G ⊙ S2 . Thus the graph G ⊙ S2 is a
F -Geometric mean graph.
A F -Geometric mean labeling of G ⊙ S2 is as shown in Figure 12.
1
4
r
r
r
6
7
r
r
3
1
4
r
r
r
6
7
r
r
3
9
10
12
13
15
16
r
r
r
r
2
4
5
8
7
r
r
r
19
r
3
6
r8
10
11
r2
r5
13
18
21
1
16
17
19
r
27
23
25
29
12
r 23
r 27
r 29
31 r❍
❍
33 31 ❍
❍ r 33
r 35
37 r 35
39
9
37
42
11
21
r
r 41
r14
14
25
38
36
r
40
36
r 39
r 41
r40
r38
9
10
12
13
15
16
r
r
1
2
4
3
5
r5
8
7
13
16
17
18
r
r17
30
19
r
19
18
28
r28
24
r
r
24 P
P Pr 26
22 P
22
21
22
r
23
✟ r 20
✟
✟
r✟ 20
32
r
34
30
26
25
9
r11
31
r
33
15
29
31
r 29
r 33
r 35
35
37 r❍
❍❍
39 37 ❍ r 39
r 41
43 r 41
45
43
48 r 46
12 25 44
r14
14
r 34
r 32
15
6
r8
10
r 11
r
r
r
r2
42
r
38
47
42
40
36
r 45
r 46
r47
r44
r 40
r 38
r17
36
18
34
r34
30
r
r
30 P
P Pr 32
27 28 P
28
26
r27
26 r
r24
24
✟ r 20
✟
✟
r ✟ 20 21
r 23
r 22
r
32
Figure 12. A F -Geometric mean labeling of H graph G ⊙ S2
Theorem 2.9. Ln ⊙ K1 is a F -Geometric mean graph, for n ≥ 2.
Proof. Let V (Ln ) = {ui , vi ; 1 ≤ i ≤ n} be the vertex set of the ladder Ln and
E(Ln ) = {ui vi ; 1 ≤ i ≤ n} ∪ {ui ui+1 , vi vi+1 ; 1 ≤ i ≤ n − 1} be the edge set of the
ladder Ln . Let wi be the pendent vertex adjacent to ui , and xi be the pendent vertex
adjacent to vi , for 1 ≤ i ≤ n.
184
A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRAN
We define f : V (Ln ⊙ K1 ) → {1, 2, 3, . . . , 5n − 1} as follows:
f (u1) = 3,
f (v1 ) = 4,
f (x1 ) = 2,
f (ui ) = 5i − 3, f or 2 ≤ i ≤ n,
f (vi ) = 5i − 2, f or 2 ≤ i ≤ n,
f (wi ) = 5i − 4, f or 1 ≤ i ≤ n
and
f (xi ) = 5i − 1, f or 2 ≤ i ≤ n.
The induced edge labeling is as follows
f ∗ (u1 v1 ) = 3,
f ∗ (v1 x1 ) = 2,
f ∗ (ui vi ) = 5i − 3, f or 2 ≤ i ≤ n,
f ∗ (wi ui ) = 5i − 4, f or 1 ≤ i ≤ n,
f ∗ (vi xi ) = 5i − 2, f or 2 ≤ i ≤ n,
f ∗ (ui ui+1 ) = 5i − 1, f or 1 ≤ i ≤ n − 1,
f ∗ (vi vi+1 ) = 5i, f or 1 ≤ i ≤ n − 1.
Hence f is a F -Geometric mean labeling of Ln ⊙ K1 . Thus the graph Ln ⊙ K1 is a
F -Geometric mean graph, for n ≥ 2.
A F -Geometric mean labeling of L8 ⊙ K1 is as shown in Figure 13.
1
r6
r
1
r
2
2
11
6
3 r
3
4
r11
r
r16
16
1719
1214
r
r
r21
r26
21
26
22
r 24
r27 29 3234
r
r37
r31
31
r36
36
4
r7
5
27
22
37
32
17
12
15
30
25
20
10
35
r
r38
r
r
r
r
r
23
33
8
28
13
18
38
33
23
28
18
13
8
9
7
r9
r14
r19
r24
r29
r34
r39
Figure 13. A F -Geometric mean labeling of L8 ⊙ K1
Theorem 2.10. Ln ⊙ S2 is a F -Geometric mean graph, for n ≥ 2.
Proof. Let V (Ln ) = {ui , vi ; 1 ≤ i ≤ n} be the vertex set of the ladder Ln and
E(Ln ) = {ui vi ; 1 ≤ i ≤ n} ∪ {ui ui+1 , vi vi+1 ; 1 ≤ i ≤ n − 1} be the edge set of the
ladder Ln .
F -GEOMETRIC MEAN LABELING OF SOME CHAIN GRAPHS AND THORN GRAPHS 185
(i)
(i)
(i)
Let w1 and w2 be the pendant vertices at each vertex ui , for 1 ≤ i ≤ n and x1
(i)
and x2 be the pendant vertices at each vertex vi , for 1 ≤ i ≤ n. Therefore
(i)
(i)
(i)
(i)
V (Ln ⊙ S2 ) = V (Ln ) ∪ {w1 , w2 , x1 , x2 ; 1 ≤ i ≤ n}
and
(i)
(i)
(i)
(i)
E(Ln ⊙ S2 ) = E(Ln ) ∪ {ui w1 , ui w2 , vi x1 , vi x2 ; 1 ≤ i ≤ n}.
We define f : V (Ln ⊙ S2 ) → {1, 2, 3, . . . , 7n − 1} as follows :

i=1
 3
7i − 2
2 ≤ i ≤ n and i is even
f (ui ) =
 7i − 5
3 ≤ i ≤ n and i is odd,

i=1
 5
7i − 4
2 ≤ i ≤ n and i is even
f (vi ) =

7i − 1
3 ≤ i ≤ n and i is odd,
7i − 3
1 ≤ i ≤ n and i is even
(i)
f (w1 ) =
7i − 6
1 ≤ i ≤ n and i is odd,

i=1
 2
(i)
7i − 1
2 ≤ i ≤ n and i is even
f (w2 ) =

7i − 4
3 ≤ i ≤ n and i is odd,
7i − 6
1 ≤ i ≤ n and i is even
(i)
f (x1 ) =
7i − 3
1 ≤ i ≤ n and i is odd,

i=1
 6
(i)
7i − 5
2 ≤ i ≤ n and i is even
f (x2 ) =
 7i − 2
3 ≤ i ≤ n and i is odd.
The induced edge labeling is as follows:
f ∗ (ui ui+1 ) = 7i − 1, f or 1 ≤ i ≤ n − 1,
f ∗ (ui vi ) = 7i − 4, f or 1 ≤ i ≤ n,
f ∗ (vi vi+1 ) = 7i, f or 1 ≤ i ≤ n − 1,
7i − 3
1≤i≤n
(i)
∗
f (ui w1 ) =
7i − 6
1≤i≤n
7i − 2
1≤i≤n
(i)
f ∗ (ui w2 ) =
7i − 5
1≤i≤n
7i − 6
1≤i≤n
(i)
f ∗ (vi x1 ) =
7i − 3
1≤i≤n
and
f
∗
(i)
(vi x2 )
=
7i − 5
7i − 2
and i is even
and i is odd,
and i is even
and i is odd,
and i is even
and i is odd,
1 ≤ i ≤ n and i is even
1 ≤ i ≤ n and i is odd.
186
A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRAN
Hence, f is a F -Geometric mean labeling of Ln ⊙ S2 . Thus the graph Ln ⊙ S2 is a
F -Geometric mean graph, for n ≥ 2.
A F -Geometric mean labeling of L7 ⊙ S2 is as shown in Figure 14.
1
1
r
2 r 11
r
3
r
2 11
6
3
5
4
7
r
5
8
13
r
r 15
17r 25
r
27
r
r 29
41 43
r
31
r 39r
r
30 39
12 15 16 25 26 29
13 r 20 r 27 r 34
12
26
30
16
10
17
31
24
r
r10 14 r20 21 24
r
38
r 42
r
9 18
19 22
28
34
r 35
23 32☞ 33 36 ✂
r
r r
r r
r r
r ☞r☞
r ✂r✂
4
6 8
9 18
19 22
23 32
33 36
45
r
40 43 44
41 r
44
40
45
38
r48
❏
47
r r ❏❏r
37 46
37 46
47
Figure 14. A F -Geometric mean labeling of L7 ⊙ S2
References
[1] A. Durai Baskar, S. Arockiaraj and B. Rajendran, Geometric Mean Graphs, (Communicated).
[2] J. A. Gallian, A dynamic survey of graph labeling, The Electronic Journal of Combinatorics,
17(2011).
[3] F. Harary, Graph theory, Addison Wesely, Reading Mass., 1972.
[4] R. Ponraj and S. Somasundaram, Further results on mean graphs, Proceedings of Sacoeference,
(2005), 443–448.
[5] A. Rosa, On certain valuation of the vertices of graph, International Symposium, Rome, July
1966, Gordon and Breach, N. Y. and Dunod Paris (1967), 349–355.
[6] S. Somasundaram and R. Ponraj, Mean labeling of graphs, National Academy Science Letter,
26(2003), 210–213.
[7] S. Somasundaram and R. Ponraj, Some results on mean graphs, Pure and Applied Mathematika
Sciences, 58(2003), 29–35.
[8] S. Somasundaram, P. Vidhyarani and R. Ponraj, Geometric mean labeling of graphs, Bulletin
of Pure and Applied sciences, 30E (2011), 153–160
[9] S. Somasundaram, P. Vidhyarani and S. S. Sandhya, Some results on Geometric mean graphs,
International Mathematical Form, 7 (2012), 1381–1391.
[10] R. Vasuki and A. Nagarajan, Further results on mean graphs, Scientia Magna, 6(3) (2010),
1–14.
Department of Mathematics,
Mepco Schlenk Engineering College,
Mepco Engineering College (Po) - 626 005
Sivakasi, Tamilnadu, INDIA.
E-mail address: [email protected], sarockiaraj [email protected],
[email protected]