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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) vol.36 no.2 Antofagasta jun. 2017

http://dx.doi.org/10.4067/S0716-09172017000200283 

Articles

Some new classes of vertex-mean graphs

A. Lourdusamy1 

M. Seenivasan2 

1 St. Xavier's College (Autonomous), India , e-mail : lourdugnanam@hotmail.com

2 Sri Paramakalyani College, India, e-mail : msvasan_22@yahoo.com

Abstract:

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 edgesin G that are incident at v, N denotes the set of all natural numbers and Round is the it nearest integer function. A graph that has a vertex-mean labeling is called vertex-mean graph or V - mean graph. In this paper, we study V - mean behaviour of certain new classes of graphs and present a method to construct disconnected V - mean graphs.

Keywords:  Mean labeling; edge labeling; vertex-mean labeling; vertex-mean graphs

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 cite1 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 v1v p , v1v 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 ,

6. The graph ,

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.

Figure 1 V - mean labeling of S(K 1,5 ) and S(K 1,6

case 1: n is odd.

Let . Define as follows:

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

Table 1. Induced vertex labels

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.

Figure 2 . V - mean labelings of dragons.  

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 u1, u2, 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

and

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.

Figure 3 V - mean labeling of G obtained from K 1,5 and K 1,6  

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 v1, v2, v2, v3, v1, v3 and v n v1. The odd and even integers of are respectively arranged in increasing sequences α 1 2 ,...,α r−2 and β12,...,βn−r−1 and αk is assigned to vk+2vk+3,and βk .

Table 2. Induced vertex labels

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 C8 (3,1) and C9 (3,1) are shown in Figure 4.

Figure 4 𝑉−mean labeling of 𝐶 𝑛 (3,1) for 𝑛=8,9.  

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.

Figure 5 A V - mean labeling of a graph obtained from C 7 and C 9

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 vi+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.

Figure 6 A V - mean labeling of C 8 ∪ C 12  

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 : v1v2v3v1 and Cn: v3v4…vn+2v3. Let . Define as follows: The integers 0, 1, 2, 3, 4 are assigned respectively to the edges v1v2,v3v1,vn+2v3,v2v3,v3v4. The odd and even integers of {5,6,7,…, n+2} are arranged respectively in increasing sequences and is assigned to vk+3vk+4, and is assigned to vn+2-kvn+3-k .

Table 3. Induced vertex labels

Table 4. Induced vertex labels

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 C4: v1v2v3v4v1 and Cn: v4v5… vn+3v4. Let . Define as follows: The integers 0, 1, 2, 3, 4, 5, 6 are assigned respectively to the edges v1v2, v2v3, v3v4, vn+3v4, v4v5, v4v1, v5v6. The odd and even integers of {7,8,…,n + 3} are arranged respectively in increasing sequences and and is assigned to vk+5vk+6, and is assigned to vn+3-kvn+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 C3 and C8 and C4 and C8 𝐶 8 are shown in Figure 7.

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.

Table 5. V -mean graphs

Let f be a V - mean labeling of G(p1,q1) and g be a V - mean labeling of H(p2,q2). Observe that the graph G ∪ H has order p = p1 + p2 and size q = q1 + q2. 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(p1, q1) is a V - mean graph of type -A and H(p2, q2) is a V - mean graph of type -B, then G ∪ H is V - mean.

Theorem 3.5. If G(p1, q1) is a V - mean graph of type -A and H(p2, q2) is a V - mean graph of type -AB, then G ∪ H is V - mean graph of type -A.

Theorem 3.6. If G(p1, q1) is a V - mean graph of type-AB and H(p2, q2) is a V - mean graph of type-B, then G ∪ H is V - mean graph of type-B.

Theorem 3.7. If both G(p1, q1) and H(p2, q2) 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(p1, q1) and H(p2, q2) are V -mean graphs of type-AB, then the graph mG ∪ nH is mean graph of type-AB.

Corollary 3.11. If G(p1, q1) is a V -mean graph of type-A and H(p2, q2) 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 mPn where , the graph , where , the graph Cn ∪ kPm where , the graph , where , the graph where are some of such graphs. To illustrate this a V - mean labeling of C10 ∪ P4 ∪ P5 and a V -mean labeling of (C8 ∪ C12) ∪ K1,8 are given in Figure 8 and Figure 9 respectively.

References

[1] B. D. Acharya and K. A. Germina, Vertex-graceful graphs, Journal of Discrete Mathematical Science & Cryptography, Vol. 13, No. 5, pp. 453-463, (2010). [ Links ]

[2] J. A. Gallian. A dynamic survey of graph labeling. The electronic journal of combinatorics, 18, (2011) \#DS6. [ Links ]

[3] A. Lourdusamy and M. Seenivasan. Vertex-mean graphs. International journal of Mathematical Combinatorics, 3, pp. 114-120, (2011). [ Links ]

[4] S. Somasundaram and R. Ponraj, Mean labelings of graphs, Natl Acad, Sci. Let., 26, pp. 210-213, (2003). [ Links ]

Received: April 2016; Accepted: January 2017

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