1. Introduction

A vertex labeling of a graph *G* is an assignment *f* of labels to the vertices of *G* that induces a label for each edge *xy* depending on the vertex labels. An *edge labeling* of a graph *G* is an assignment *f* of labels to the edges of *G* that induces a label for each vertex *v* depending on the labels of the edges incident on it. Vertex labelings such as *graceful labeling, harmonious labeling* and labeling and edge labelings such *as edge-magic labeling, (a,d)-anti magic labeling* and *vertex-graceful labeling* are some of the interesting labelings found in the dynamic survey of graph labeling by Gallian ^{2} In fact Acharya and Germina cite^{1} has introduced *vertex-graceful graphs*, as an edge-analogue of *graceful graphs*.

A *mean labeling f* is an injective function from *V* to the set
such that the set of edge labels defined by the rule
for each edge
. The *mean labeling* was introduced by Somasundaram and Ponraj [^{4}]. Observe that, in a variety of practical problems, the arithmetic mean, *X* of a finite set of real numbers
serves as a better estimate for it, in the sense that
is zero and
is the minimum. If it is required to use a single integer in the place of *X* then *Round(X)* does this best, in the sense that
and
are minimum, where *Round (Y),*
*nearest integer function* of a real number, gives the integer closest to *(Y)*, to avoid ambiguity, it is defined to be the nearest integer greater than *Y*, if the fraction of y is 0.5. Motivated by this and the concept of *vertex-graceful graphs*,Lourdusamy and Seenivasan ^{3} introduced *vertex-mean labeling* as an edge analogue of *mean labeling* as follows:

A *vertex-mean labeling* of a *(p, q)* graph *G = (V, E)* is defined as an injective function
such that the function
defined by the rule
satisfies the property that
, where _{
Ev
} denotes the set of edges in *G* that are incident at *v* and **N** denotes the set of all natural numbers. A graph that has a vertex-mean labeling is called *vertex-mean graph* or *V* - mean graph. They, obtained necessary conditions for a graph to be a *V* - mean graph, and proved that any 3-regular graph of order
is not a *V* - mean graph. They also proved that the path _{
Pn
} , where
and the cycle _{
Cn
} , the Corona
, where
and
the star graph _{
K1,n
} if and only if
, and the crown
are *V* - mean graphs. A *dragon* is a graph obtained by identifying an end point of a path _{
Pm
} with a vertex of the cycle _{
Cn
} and _{
mPn
} denotes the disjoint union of m copies of the path P_{
n
} . For
, denotes the graph obtained from the cycle C_{
n
} with consecutive vertices v_{
1
} , v_{
2
} , …, v_{
n
} by adding the r chords v_{1}v_{
p
} , v_{1}v_{
p+1
} ,…, v_{
1
} v_{
p+r-1
} . In this paper we present the V - mean labeling of the following graphs:

1. The graph S(K_{
1,n
} ), obtained by subdividing every edge of K_{
1,n
} ,

2. Dragon graph,

3. The graph obtained by identifying one vertex of the cycle C_{
3
} 𝐶 3 with the central vertex of K_{
1,n
} ,

4. The graph C_{
n
} (3,1),

5. The graph obtained from the two cycles C_{
n
} and C_{
m
} by adding a new edge joining a vertex of C_{
n
} and C_{
m
} where
,

7. The graph obtained by identifying one vertex of the cycle C_{
m
} with a vertex of C_{
n
} when m = 3 or 4.

We also explain a method to obtain disconnected V - mean graphs from V - mean graphs. Following are some of the graphs so obtained: the graph
, the graph mP_{
n
} where
, the graph
, where
, the graph
where
, the graph
, where
, the graph
where
.

2. New classes of V-mean graphs

**Theorem 2.1.**The graph S(K_{
1,n
} ), obtained by subdividing every edge of K_{
1,n
} , exactly once, is a V - mean graph.

**Proof**. Let
and
be the vertex set and edge set of S(K_{
1,n
} ) respectively. Then G has order 2n + 1 and size 2n.

**case 1:** n is odd.

Then it is easy to verify that

Hence

**case 2:** n is even.

Let n = 2m. Define as follows:

Then, it is easy to verify that

Hence

Thus, S(K_{
1,n
} ) is V - mean. V - mean labeling of S(K_{
1,5
} ) and S(K_{
1,6
} ) are shown in Figure 1.

**Theorem 2.2.** A dragon graph is V - mean.

**Proof.** Let 𝐺 be a dragon consisting of the path P_{
m
}: v_{
1
} v_{
2
} …v_{
m
} and the cycle C_{
n
}: u_{
1
} u_{
2
} …u_{
n
} . Let v_{
m
} be identified with u_{
n
} and
. Let
and
be the edges of 𝐺. Observe that G has order and size both equal to m + n - 1. The edges of G are labeled as follows:

For , the integer is assigned to the edge . The odd and even integers from 1 to n are respectively arranged in increasing sequences and and is assigned to and is assigned to

Clearly the edges of G receive distinct labels from and the vertex labels induced are 1,2,…, m + n - 1 as illustrated in Table 1.

1. Thus G is V - mean.

For example V - mean labelings of dragons obtained from P_{
5
} and C_{
7
} and P_{
6
} and C_{
7
} are shown in Figure 2.

**Theorem 2.3.** Let G be a graph obtained by identifying one vertex of the cycle C_{
3
} with the central vertex of K_{
1,n
} . Then G is V - mean.

**Proof.** Let u_{1}, u_{2}, u_{
3
} be the consecutive vertices of C_{
3
} . Let
with deg w = n and u_{
1
} be identified with w. Then G is of order and size both equal to n + 3. Let
. Define
as follows:

Then, it follows easily that

Hence
. Thus G is a V - mean graph. V - mean labeling of the graphs obtained from K_{
1,5
} and K_{
1,6
} by idenntifying the central vertex of each with a vertex of C_{
3
} as shown in Figure 3.

**Theorem 2.4.**The graph C_{
n
} (3.1) is a V - mean graph.

Proof.Let G = C_{
n
} (3,1). Let
and
Then G has order n and size n + 1. Let
. The edges of G are assigned labels as follows: Define
as follows : The integers 0, 1, 2, 3 are respectively assigned to the edges v_{1}, v_{2}, v_{2}, v_{3}, v_{1}, v_{3} and v_{
n
} v_{1}. The odd and even integers of
are respectively arranged in increasing sequences α_{
1
} ,α_{
2
} ,...,α_{
r−2
} and β_{1},β_{2},...,β_{n−r−1} and αk is assigned to v_{k}+_{2}v_{k+3},and β_{k} .

Clearly the assignment is an injective function and the set of induced vertex labels is {1,2,…,n}, as illustrated in Table 2. Thus G is a V - mean graph. V - mean labeling of C_{8} (3,1) and C_{9} (3,1) are shown in Figure 4.

**Theorem 2.5.**
, the graph obtained from the two cycles C_{
n
} and C_{
m
} by adding a new edge joining a vertex of C_{
n
} and C_{
m
} is a V - mean graph.

**Proof.** Let G be the graph consisting of two cycles C_{
n
}: v_{
1
} v_{
2
} …v_{
n
} and C_{
m
}: u_{
1
} u_{
2
} …v_{
m
} and e_{
0
} =v_{
n
} u_{
1
} be the bridge connecting them. The G n has order m + n and size m + n + 1. Let e_{
i
} = v_{
i
} v_{
i+1
} ,
and
. Let
. Define
as follows:

Then

Clearly f is an injective function and the set of induced vertex labels is {1,2,…,n + m}. Hence the theorem.

A V - mean labeling of the graph obtained from C_{
7
} and C_{
9
} is shown in Figure 5.

**Theorem 2.6.**The graph C_{
n
} ∪ C_{
m
} is a V - mean graph.

**Proof.** Let {e_{
1
} , e_{
2
} , …, e_{
n
} } be the edge set of C_{
n
} such that e_{
i
} = v_{
i
} v_{i+1},
and
be the edge set of C_{
m
} such that
. Then the graph G = C_{
n
} U C_{
m
} has order and size both equal to m + n. Let
Define
as follows:

Then

Clearly f is an injective function and the set of induced vertex labels is {1,2,…, n + m}. Hence the theorem.

A V - mean labeling of C_{
8
} ∪ C_{
12
} is shown in Figure 6.

**Theorem 2.7.** If
, the graph G obtained by identifying one vertex of the cycle C_{
m
} with a vertex of C_{
n
} is a V - mean graph.

**Proof**

**case 1** m = 3

Let G be the graph consisting of two cycles C_{
3
}: v_{1}v_{2}v_{3}v_{1} and C_{n}: v_{3}v_{4}…v_{n+2}v_{3}. Let
. Define
as follows: The integers 0, 1, 2, 3, 4 are assigned respectively to the edges v_{1}v_{2},v_{3}v_{1},v_{n+2}v_{3},v_{2}v_{3},v_{3}v_{4}. The odd and even integers of {5,6,7,…, n+2} are arranged respectively in increasing sequences
and
is assigned to v_{k+3}v_{k+4}, and
is assigned to v_{n+2-k}v_{n+3-k} .

Clearly the assignment is an injective function and the set of induced vertex labels is {1,2,…, n + 2}, as illustrated in Table 3. Thus G is a V - mean graph.

**case 2** m = 4.

Let G be the graph consisting of two cycles C_{4}: v_{1}v_{2}v_{3}v_{4}v_{1} and C_{n}: v_{4}v_{5}… v_{n+3}v4. Let
. Define
as follows: The integers 0, 1, 2, 3, 4, 5, 6 are assigned respectively to the edges v_{1}v_{2}, v_{2}v_{3}, v_{3}v_{4}, v_{n+3}v_{4}, v_{4}v_{5}, v_{4}v_{1}, v_{5}v_{6}. The odd and even integers of {7,8,…,n + 3} are arranged respectively in increasing sequences
and
and
is assigned to v_{k+5}v_{k+6}, and
is assigned to v_{n+3-k}v_{n+4-k}.

Clearly the assignment is an injective function and the set of induced vertex labels is {1,2,…,n + 3}, as illustrated in Table 4. Thus G is a V - mean graph.

Fox example V - mean labeling of graphs obtained from C_{3} and C_{8} and C_{4} and C_{8} 𝐶 8 are shown in Figure 7.

3. Some disconnected V -mean graphs

In this section we present a method to construct disconnected V - mean graphs from V - mean graphs. The following observation is obvious from the definition of V - mean labeling.

**Observation 3.1.** If f is any V - mean labeling of a (p, q) graph G, then
for some edge
. In particular, if
then
for every edge
and hence
for some edge
.

**Notation 3.2.** We call a V -mean labeling f of a graph G(p,q) as type-A, if
for every edge
for every edge
for every edge
, we call G as V - mean graph of type -S if it has a V - mean labeling f of type -S.

**Remark 3.3.** We observe that the V - mean graphs presented in ^{3} and Theorem 2.1 through Theorem 2.7 can be classified as given in Table 5.

Let f be a V - mean labeling of G(p_{1},q_{1}) and g be a V - mean labeling of H(p_{2},q_{2}). Observe that the graph G ∪ H has order p = p_{1} + p_{2} and size q = q_{1} + q_{2}. Define
as follows:

Suppose f is of type-A and g is of type-B. Then for every edge and for every edge . As f and g are injective functions, for every edge and for every edge is injective.

Suppose f is of type -A and g is of type-AB. Then for every edge and for every edge . As f and g are injective functions, for every edge and for every edge , h is injective and for every edge .

Suppose f is of type- AB and g is of type- B. Then for every edge and for every edge . As f and g are injective functions, for every edge and for every edge is injective and for every edge .

Suppose, both f and g are of type -AB. Then for every edge and for every edge . As, f and g are injective functions, for every edge and for every edge is injective and for every edge .

The set of induced vertex labels of G U H in all four cases is as follows:

Thus we have the following four theorems.

**Theorem 3.4.** If G(p_{1}, q_{1}) is a V - mean graph of type -A and H(p_{2}, q_{2}) is a V - mean graph of type -B, then G ∪ H is V - mean.

**Theorem 3.5.** If G(p_{1}, q_{1}) is a V - mean graph of type -A and H(p_{2}, q_{2}) is a V - mean graph of type -AB, then G ∪ H is V - mean graph of type -A.

**Theorem 3.6.** If G(p_{1}, q_{1}) is a V - mean graph of type-AB and H(p_{2}, q_{2}) is a V - mean graph of type-B, then G ∪ H is V - mean graph of type-B.

**Theorem 3.7.** If both G(p_{1}, q_{1}) and H(p_{2}, q_{2}) are V - mean graphs of type -AB then the graph G ∪ H is V - mean graph of type -AB.

**Corollary 3.8.** Let G be a tree or a unicyclic graph or a two regular graph. If G is V -mean and H is a V -mean graph of type-B, then G ∪ H is V - mean.

**Corollary 3.9.** If G(p, q) is V- mean graph of type-AB then, the graph mG is V -mean graph type-AB.

**Corollary 3.10.** If both G(p_{1}, q_{1}) and H(p_{2}, q_{2}) are V -mean graphs of type-AB, then the graph mG ∪ nH is mean graph of type-AB.

**Corollary 3.11.** If G(p_{1}, q_{1}) is a V -mean graph of type-A and H(p_{2}, q_{2}) is a V - mean graph of type-AB then G ∪ H mH is V -mean graph of type-A.

It is interesting to note that a number of disconnected V - mean graphs can be obtained by applying Theorem 3.4 through Corollary 3.11 on V - mean graphs listed in Table 5. For example, the graph
, where
, the graph mP_{n} where
, the graph
, where
, the graph C_{n} ∪ kP_{m} where
, the graph
, where
, the graph
where
are some of such graphs. To illustrate this a V - mean labeling of C_{10} ∪ P_{4} ∪ P_{5} and a V -mean labeling of (C_{8} ∪ C_{12}) ∪ K_{1,8} are given in Figure 8 and Figure 9 respectively.