1. Introduction
The graphs considered here will be finite, undirected and simple. 
and 
will denote the vertex set and edge set of a graph G. The cardinality of the vertex set of a graph G is denoted by p and the cardinality of its edge set is denoted by q. The corona 
of two graphs 
and 
is defined as the graph obtained by taking one copy of 
(with 
vertices) and 
copies of 
and then joining the ith vertex of 
to all the vertices in the ith copy of
. If e = uv is an edge of G and w is a vertex not in G then e is said to be subdivided when it is replaced by the edges uw and wv. The graph obtained by subdividing each edge of a graph G is called the subdivision graph of G and it is denoted by
. The graph 
is called the ladder. A dragon is a graph formed by joining an end vertex of a path 
to a vertex of the cycle
. It is denoted as
. The triangular snake 
is obtained from the path 
by replacing every edge of a path by a triangle
. The quadrilateral snake 
is obtained from the path 
by every edge of a path is replaced by a cycle
. The concept of pair sum labeling has been introduced in [1]. The Pair sum labeling behavior of some standard graphs like complete graph, cycle, path, bistar, and some more standard graphs are investigated in [1-3]. That all the trees of order ≤9 are pair sum have been proved in [4]. Terms not defined here are used in the sense of Harary [5]. Let x be any real number. Then 
stands for the largest integer less than or equal to x and 
stands for the smallest integer greater than or equal to x. Here we investigate the pair sum labeling behavior of
, for some standard graphs G.
2. Pair Sum Labeling
Definition 2.1. Let G be a 
graph. An injective map 
is called a pair sum labeling if the induced edge function, 
defined by 
is one-one and 
is either of the form
or
according as q is even or odd. A graph with a pair sum labeling defined on it is called a pair sum graph.
Theorem 2.2 [1]. Any path is a pair sum graph.
Theorem 2.3 [1]. Any cycle is a pair sum graph.
3. On Standard Graphs
Here we investigate pair sum labeling behavior of 
and
.
Theorem 3.1. If n is even, 
is a pair sum graph.
Proof. Let 
be the cycle 
and let 
be the path
.
Case 1. 
Define
by
.
Here
Therefore f is a pair sum labeling.
Case 2. 
Define
by
Here
Hence f is a pair sum labeling.
Case 3. 
Label the vertex
, 
as in Case 1. Then label 
to
.
Case 4. 
Assign the label 
to 
and assign the label to the remaining vertices as in Case 2.
Illustration 1. A pair sum labeling of 
is shown in Figure 1.
Theorem 3.2. 
is pair sum graph if n is even.
Proof: Let 
be the vertices of 
and u, v, w, z. be the vertices in 
Let
and
Define
by
Here
Therefore f is a pair sum labeling.
Illustration 2. A pair sum labeling of 
is shown in Figure 2.
4. On Subdivision Graph
Here we investigate the pair sum labeling behavior of 
for some standard graphs G.
Theorem 4.1. 
is a pair sum graph, where 
is a ladder on n vertices.
Proof. Let
Let
Case 1: n is even.
When n = 2, the proof follows from the Theorem 2.3. For n > 2Define
by
When n = 4,
For n > 4,
Therefore f is a pair sum labeling.
Case 2. n is odd.
Clearly 
and hence 
is a pair sum graph by Theorem 2.2. For n > 1Define
by
Therefore
and
when n > 5,
Then f is a pair sum labeling.
Illustration 3. A pair sum labeling of 
is shown in Figure 3.
Theorem 4.2. 
is a pair sum graph Proof. Let
Let
Case 1. n is even.
Define
by
Here
Then f is pair sum labeling.
Case 2. n is odd.
Define
by
Here
Then f is pair sum labeling.
Illustration 4. A pair sum labeling of 
is shown in Figure 4.
Theorem 4.3. 
is a pair sum graph.
Proof. Let
Let
Case 1. n is even.
When n = 2, the proof follows from Theorem 2.2. For n > 2, Define
by
Here
For n > 4,
Then f is pair sum labeling.
Case 2. n is odd.
Since
, which is a pair sum graph by Theorem 2.3. For n > 1, Define
by
Here
When n > 5,
Then f is pair sum labeling.
Illustration 5. A pair sum labeling of 
is shown in Figure 5.
Theorem 4.4. 
is a pair sum graph where 
is a triangular snake with n triangle.
Proof. Let
and
Case 1. n is even.
When n = 2, Define f(u1) = 7, f(u2) = 6, f(u3) = 1, f(u4) = –6, f(u5) = –7, f(v1) = 5, f(v2) = 2, f(v3) = –4, f(v4) = –5, f(w1) = 3, f(w2) = –3. When n > 2, Define
by
For n = 4,
For n > 4
Then f is pair sum labeling.
Case 2. n is odd.
Clearly
, and hence 
is a pair sum graph by Theorem 2.3.
For > 1, Define
by
,
Here n = 3,
For n > 3,
Then f is pair sum labeling.
Illustration 6. A pair sum labeling of 
is shown in Figure 6.
Theorem 4.5. 
is a pair sum graph.
Proof. Let
and
Case 1. n is even.
When n = 2, Define f(u1) = 11, f(u2) = 6, f(u3) = 1, f(u4) = –6, f(u5) = –11, f(w1) = 9, f(w2) = 2, f(w3) = –4, f(w4) = –9, f(v1) = 7, f(v2) = 5, f(v3) = 3, f(v4) = –3, f(v5) = –5, f(v6) = –7. When > 2, Define
by
Here
Then f is pair sum labeling.
Case 2. n is odd.
is a pair sum graph follows from Theorem 2.3.When n > 1. Define
by
For n = 3,
n > 3,
Then f is pair sum labeling Illustration 7. A pair sum labeling of 
is shown in Figure 7.
5. Acknowledgements
We thank the referees for their valuable comments and suggestions.