1. Introduction
We begin with simple, finite, connected and undirected graph 
with
vertices and 
edges. For standard terminology and notations we follow Gross and Yellen [1]. We will provide brief summary of definitions and other information which are necessary for the present investigations.
Definition 1.1 If the vertices are assigned values subject to certain condition(s) then it is known as graph labeling.
Any graph labeling will have the following three common characteristics:
1) A set of numbers from which vertex labels are chosen;
2) A rule that assigns a value to each edge;
3) A condition that this value has to satisfy.
According to Beineke and Hegde [2] graph labeling serves as a frontier between number theory and structure of graphs. For a dynamic survey of various graph labeling problems along with extensive bibliography we refer to Gallian [3].
Definition 1.2 A mapping 
is called binary vertex labeling of 
and 
is called the label of the vertex 
of 
under
.
Notation 1.3 If for an edge
, the induced edge labeling 
is given by
. Then
Definition 1.4 A binary vertex labeling 
of a graph 
is called a cordial labeling if 
and
. A graph 
is cordial if it admits cordial labeling.
The concept of cordial labeling was introduced by Cahit [4]. Some labeling schemes are also introduced with minor variations in cordial theme. Some of them are product cordial labeling, total product cordial labeling and prime cordial labeling. The present work is focused on prime cordial labeling.
Definition 1.5 A prime cordial labeling of a graph 
with vertex set 
is a bijection 
and the induced function 
is defined by
satisfies the condition 
A graph which admits prime cordial labeling is called a prime cordial graph.
The concept of prime cordial labeling was introduced by Sundaram [5] et al. and in the same paper they have investigated several results on prime cordial labeling. Vaidya and Vihol [6] have also discussed prime cordial labeling in the context of graph operations while in [7] the same authors have discussed prime cordial labeling for some cycle related graphs. Vaidya and Shah [8] have investigated many results on this concept. In the present paper we obtain some new prime cordial graphs.
Definition 1.6 Bistar is the graph obtained by joining the apex vertices of two copies of star
.
Definition 1.7 For a graph G the split graph is obtained by adding to each vertex v a new vertex 
such that 
is adjacent to every vertex that is adjacent to v in G. The resultant graph is denoted as
.
Definition 1.8 For a simple connected graph G the square of graph G is denoted by 
and defined as the graph with the same vertex set as of G and two vertices are adjacent in 
if they are at a distance 1 or 2 apart in G.
Definition 1.9 The middle graph M(G) of a graph G is the graph whose vertex set is 
and in which two vertices are adjacent if and only if either they are adjacent edges of G or one is a vertex of G and the other is an edge incident with it.
2. Main Results
Theorem 2.1 
is a prime cordial graph.
Proof: Let 
be the pendant vertices, 
be the apex vertex of 
and 
are the vertices corresponding to 
in
. Denoting 
then 
and 
.
To define
, we consider following two cases.
Case 1: n = 2, 3 The graphs 
and 
are to be dealt separately and their prime cordial labeling is shown in Figure 1.
Case 2: 
In view of the labeling pattern defined above we have 
(for n even) and
(for n odd). Thus we have
.
Hence 
is a prime cordial graph.
Illustration 2.2 Prime cordial labeling for 
is shown in Figure 2.
Figure 1.
,
and their prime cordial labelling.
Figure 2. 
and its prime cordial labelling.
Theorem 2.3 
is a prime cordial graph.
Proof: Consider 
with vertex set
where 
are pendant vertices. In order to obtain 
add 
vertices corresponding to 
where
. If 
then 
and 
. We define vertex labeling 
as follows.
In view of pattern defined above we have 
.
That is,
.
Hence 
is a prime cordial graph.
Illustration 2.4 Prime cordial labeling of the graph 
is shown in Figure 3.
Theorem 2.5 
is a prime cordial graph.
Proof: Consider 
with vertex set 
where 
are pendant vertices. Let G be the graph 
then 
and 
.
To define
, we consider following two cases.
Case 1: 
The graphs 
and 
are to be dealt separately and their prime cordial labeling is shown in Figure 4.
Case 2: 
Choose a prime number p such that
,
, where 
is distinct even numbers between 4 and 2n + 2 except 2p with
.
,
for
.
In view of the above defined labeling pattern we have
.
Thus in both the cases we have
.
Hence 
is a prime cordial graph.
Illustration 2.6 Prime cordial labeling of the graph 
is shown in Figure 5.
Figure 3. 
and its prime cordial labelling.
Figure 4.
, 
and their prime cordial labelling.
Theorem 2.7 
is not a prime cordial graph for 
= 2, 3.
Proof: Let 
and 
are respectively be the vertices and edges of
. Add vertices 
corresponding to the edges 
in order to obtain middle graph of
. Let 
be the graph
. Then 
and
.
For the graph 
the possible assignment of labels to adjacent vertices are (1,2), (1,3), (2,3). Such assignment will generate no edge with label 0 and two edges with label 1. That is,
. Therefore 
is not a prime cordial graph.
For the graph 
the possible assignment of labels to adjacent vertices are (1,2),(1,3), (1,4), (1,5), (2,3), (2,4), (2,5), (3,4), (3,5). Such assignment will generate maximum one edge with label 0 and minimum four edges with label 1. That is,
. Therefore 
is not a prime cordial graph.
Hence 
is not a prime cordial graph for 
Theorem 2.8 
is a prime cordial graph for 
Proof: Let 
and 
are respectively be the vertices and edges of
. Add vertices 
corresponding to the edges 
in order to obtain middle graph of
. Let 
be the graph
. Then 
and 
. To define 
, we consider following two cases.
Case 1: 
is even, 
In view of the above defined labeling pattern we have
.
Case 2: 
is odd, 
Figure 5. 
and its prime cordial labelling.
,
,
In view of the above defined labeling pattern we have
Thus in both the cases we have
.
Hence 
is a prime cordial graph for 
Illustration 2.9 Prime cordial labeling of the graph 
is shown in Figure 6.
Theorem 2.10 
is not a prime cordial graph for n = 3 to n = 7.
Proof: Let 
be the apex vertex of wheel 
and 
be the rim vertices. Then 
and 
For the graph 
the possible pairs of labels of adja-
Figure 6. 
and its prime cordial labelling.
cent vertices will be (1,2), (1,3), (1,4), (2,3), (2,4), (3,4). Such assignment will generate maximum one edge with label 0 and minimum five edges with label 1. That is,
. Hence 
is not a prime cordial graph.
For the graph 
the possible assignment of labels to adjacent vertices will be (1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (2,5), (3,4), (3,5), (4,5). Such assignment will generate maximum one edge with label 0 and minimum seven edges with label 1. That is,
. Hence 
is not a prime cordial graph.
For the graph 
the possible assignment of labels to adjacent vertices will be (1,2), (1,3), (1,4), (1,5), (1,6), (2,3), (2,4), (2,5), (2,6), (3,4), (3,5), (3,6), (4,5), (4,6), (5,6). Such assignment will generate maximum four edges with label 0 and minimum six edges with label 1. That is,
. Hence 
is not a prime cordial graph.
For the graph 
the possible assignment of labels to adjacent vertices will be (1,2), (1,3), (1,4), (1,5), (1,6), (1,7), (2,3), (2,4), (2,5), (2,6), (2,7), (3,4), (3,5), (3,6), (3,7), (4,5), (4,6), (4,7), (5,6), (5,7), (6,7). Such assignment will generate maximum four edges with label 0 and minimum eight edges with label 1. That is, 
. Hence 
is not a prime cordial graph.
Now in 
to satisfy the edge condition for prime cordial labeling it is essential to label seven edges with label 0 and seven edges with label 1 out of fourteen edges. But all the possible assignments of vertex labels will give rise to 0 labels for at most six edges and 1 labels for at least eight edges. That is,
. Hence 
is not a prime cordial graph.
Hence Wn is not a prime cordial graph for n = 3 to n = 7.
Theorem 2.11 
is a prime cordial graph for
.
Proof: Let 
be the apex vertex of wheel 
and 
be the rim vertices. To define 
, we consider the following three cases.
Case 1: n = 8,9,10 The graphs
, 
and 
are to be dealt separately and their prime cordial labeling is shown Figure 7.
Case 2: 
is even, 
In view of the above defined labeling pattern we have 
Case 3: 
is odd, 
In view of the above defined labeling pattern we have 
Thus in all the cases we have
.
Hence 
is a prime cordial graph for
.
Illustration 2.12 Prime cordial labeling of the graph 
is shown in Figure 8.
3. Concluding Remarks
As not every graph admit prime cordial labeling it is very interesting to investigate graph or graph families which admit prime cordial labeling. In this paper we have investigated some new prime cordial graphs. To investigate similar results for other graph families as well as in the context of different labeling problems is an open area of research.
Figure 8. 
and its prime cordial labelling.