We begin with simple, finite, undirected and non-trivial graph
, with vertex set 
and edge set
. Throughout this work 
denotes the cycle with 
vertices and 
denotes the path of 
vertices. In wheel 
the vertex corresponding to 
is called apex vertex and the vertices corresponding to 
are called rim vertices where
. The star 
is a graph with one vertex of degree 
called apex and 
vertices of degree one (pendant vertices). Throughout this paper 
and 
are the cardinality of vertex set and edge set respectively.
For various graph theoretic notations and terminology we follow Gross and Yellen [
Definition 1.1: If the vertices of the graph are assigned values subject to certain conditions then it is known as graph labeling.
Vast amount of literature is available in printed as well as in electronic form on different types of graph labeling. More than 1300 research papers have been published so far in last four decades. For a dynamic survey of graph labeling problems along with extensive bibliography we refer to Gallian [
Definition 1.2: A prime labeling of a graph 
is an injective function 
such that for every pair of adjacent vertices 
and
,
. The graph which admits a prime labeling is called a prime graph.
The notion of a prime labeling was originated by Entringer and was discussed in a paper by Tout et al. [
and 
are prime graphs. Lee et al [
is a prime graph if and only if 
is even. Deretsky et al. [
is a prime graph.
Definition 1.3: A vertex switching 
of a graph 
is the graph obtained by taking a vertex 
of
, removing all the edges incident to 
and adding edges joining 
to every other vertex which are not adjacent to 
in
.
Definition 1.4: A prime graph 
is said to be switching invariant if for every vertex 
of
, the graph 
obtained by switching the vertex 
in 
is also a prime graph.
Vaidya and Kanani [
corresponding to prime labeling while in the present paper we investigate further results on prime graphs.
Bertarnad’s Postulate 1.5: For every positive integer 
there is a prime 
such that
.
of order 
has at least one vertex 
(corresponding to label 1) such that 
is a prime graph.
be a prime graph of order 
with a prime labeling. If 
is the vertex corresponding to the largest prime less then or equal to 
then 
is a prime graph.
be a prime graph of order 
with a prime labeling and 
is any arbitrary vertex having prime label from the set
then 
is a prime graph.
is switching invariant.
be consecutive vertices of
. Let 
be the graph obtained by switching a vertex 
of
.
we have the following possibilities:
then in
, 
is adjacent to 
.
as follows:
, where 
is an injection.
of 
we claim that
. Because a) if 
for some 
then 
;
for some 
then 
as 
and 
are consecutive positive integers.
the proof is similar as discussed above.
for some
. Then in
, 
is adjacent to all the vertices except 
and
.
as follows:
is an injection.
of 
we claim that
.
for some 
then 
;
for some 
then 
.
for some 
then 
as 
and 
are consecutive positive integers;
for some 
then
;
under consideration admits a prime labeling. i.e. 
is a prime graph.
and the graph obtained by switching of any vertex in 
are prime graphs. Hence the result.
is shown in the 
which is not a pendant vertex is shown in the 
is switching invariant.
is the apex vertex of 
and 
is the graph obtained by switching the apex vertex 
then 
is the null graph 
on 
vertices, which does not have any edge. Then obviously it is a prime graph.
be the apex vertex and 
be the consecutive pendant vertices of
. Let 
be the graph obtained by switching the pendant vertex 
of
. So in 
every vertex 
other than 
and 
is adjacent to 
and 
only. By Bertrand’s postulate of number theory there exists at least one prime
such that 
then it is possible to define
as follows:
admits a prime labeling on
. Hence 
is a prime graph.
and the graph obtained by switching any vertex of 
are prime graphs. Hence the result.
is shown in the 
is a prime graph.
be consecutive rim vertices of 
and 
be the apex vertex of
. Let 
be the graph obtained by switching the vertex
. Thus 
and
.
as follows:
is an injection.
of 
we claim that
. 
for some 
then 
as 
and 
are consecutive positive integers.
then 
.
admits a prime labeling. i.e. 
is a prime graph.
is shown in the 
is a prime graph if 
is a prime number.
be consecutive rim vertices of 
and 
be the apex vertex of
. Let 
be the graph obtained by switching the vertex
.
as follows:
, 
and
.
is an injection.
of 
we claim that
.
for some 
then 
as 
and 
are consecutive positive integers.
for some 
then
.
for some 
then 
as 
is a positive integer less than the prime number
.
under consideration admits a prime labeling. i.e. 
is a prime graph.
is shown in the 
is not a prime graph.
be consecutive rim vertices of 
and 
be the apex vertex of
. Let 
be the graph obtained by switching the vertex
.
be a prime labeling. As 
is adjacent to four vertices 
and 
is adjacent to six vertices
, the labels of 
and 
can not be even. Moreover we have to distribute four even labels among six vertices. Therefore at least two adjacent vertices from 
will receive the even labels which contradicts the fact that 
is a prime labeling.
when 
is a composite number. However we prove a following result and pose a conjecture.
is a not a prime graph if 
is an even integer greater than 9.
be consecutive rim vertices and 
be the apex vertex of
. Let 
be the graph obtained by switching the vertex
. If possible assume that there exists a prime labeling 
on
.
the number of even integers is equal to the number of odd integers in
which is
.
is adjacent to at least 
vertices then it cannot be labeled with even integer otherwise each of its 
neighbours should receive the labels with odd integers. Consequently there should be at least 
odd integers in 
and at the most 
even integers. However, there are at least 5 even integers in
, a contradiction.
and 
have at least 
neighbours must be labeled with an odd integer. Therefore the remaining vertices 
forms a path 
in 
and these vertices will receive
odd labels and 
even labels. If
and 
denote the number of vertices with even labels and number of vertices with odd labels respectively in 
then
. Hence there must be two adjacent vertices in 
which will receive even labels which contradicts our assumption that 
is a prime labeling of
.
is a not a prime graph if 
is a composite odd integer greater than 9.