## Articulo

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## versión impresa ISSN 0716-0917

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

#### http://dx.doi.org/10.4067/S0716-09172017000200347

Articles

Skolem difference mean labeling of disconnected graphs

1 Govindammal Aditanar College for Women, India, e-mail : jeyajeyanthi@rediffmail.com

2 Dr. Sivanthi Aditanar College of Engineering, India, e-mail : selvm80@yahoo.in

3 Govenrnment Arts College for Women, India, e-mail : aymar_padma@yahoo.co.in

Abstract

Let G = (V , E) be a graph with p vertices and q edges. G is said to have skolem difference mean labeling if it is possible to label the vertices with distinct elements ƒ (x) from 1,2,2,…, p + q in such a way that for each edge and the resulting labels of the edges are distinct and are from 1,2,2,…, q. A graph that admits a skolem difference mean labeling is called a skolem difference mean graph. In this paper, we prove that the graphs graphs.

Keywords mean labeling; skolem difference mean labeling; skolem difference mean graph

1. Introduction

Throughout this paper by a graph we mean a finite, simple and undirected one. The vertex set and the edge set of a graph G are denoted by V (G) and E(G) respectively. A graph labeling is an assignment of integers to the vertices or edges or both, subject to certain conditions. Terms and notations not defined here are used in the sense of Harary ((1)). There are several types of labeling. An excellent survey of graph labeling is maintained by Gallian ((2)). The notion of mean labeling was due to Somasundaram et al.((8)). A graph G = (V ,E) with p vertices and q edges is called a mean graph if there is an injective function ƒ that maps V (G) to such that for each edge 𝒖𝒗 labeled with is odd. Then the resulting edge labels are distinct. The concept of skolem difference mean labeling was introduced by Murugan et al.((4)) and they studied the skolem difference mean labeling of H-graphs. In ((3)), they studied skolem difference mean labeling of finite union of paths. Further, the skolem difference mean labeling of were proved in ((5)). Ramya et al. ((6)), proved that , where T is a Tp - tree, caterpillar, Sm,n and were skolem difference mean graphs. In ((7)), we proved that the graphs and Bt(n,n, …n) admit skolem difference mean labeling.

We use the following definitions in the subsequent section.

Definition 1.1. A graph G = (V ,E) with p vertices and 𝑞 edges is said to have skolem difference mean labeling if it is possible to label the vertices with distinct elements f (χ) from 1,2,3,…p + q in such a way that for each edge e = 𝒖𝒗, let f* (e) = is odd and the resulting labels of the edges are distinct and are from 1,2,3. …, q. A graph that admits a skolem difference mean labeling is called a skolem difference mean graph.

Definition 1.2. Let G1 and G2 be two graphs having vertex sets V1 and V2 and edge sets E1 and E2 respectively. Then their union G1 ∪ G2 has V = V1 ∪ V2 and E = E1 ∪ E2.

Definition 1.3. A path is a walk if all the vertices and edges are distinct. A path on 𝑛 vertices is denoted by Pn The graph mPn is the disjoint union of m copies of the path Pn

Definition 1.4. The wheel graph Wn is obtained by joining a central vertex to all vertices of the cycle Cn by an edge.

Definition 1.5. The fan graph 𝐹 𝑛 is obtained by joining a vertex to all vertices of the path Pn by an

2. Main results

In this section we prove that a skolem difference mean graphs.

Theorem 2.1. The union of two cycles is a skolem difference mean graph.

Proof Case(i). m is odd.

Let m = 2l + 1.

Subcase(i). 𝑛 is odd.

Let n = 2 k + 1

For each vertex label f, the induced edge label f* is calculated as follows:

Therefore, and hence f is a skolem difference mean labeling.

Subcase(ii). n is even.

Let n = 2K ˃ 1.

For each vertex label f, the induced edge label f* is calculated as follows:

Therefore, we get the edge labels are from . Hence f is a skolem difference mean labeling.

Case(ii). m is even.

Let m = 2I.

Subcase(i). n is odd.

Let n = 2K + 1.

For each vertex label f, the induced edge label f* is calculated as follows:

Therefore, f is a skolem difference mean labeling.

Subcase(ii). n is even.

Let n = 2K ˃ 1.

Therefore, f is a skolem difference mean labeling and hence Cm ∪ Cn is a skolem difference mean graph. The skolem difference mean labeling of C7 ∪ C8 and C9 ∪ C3 are shown in Figure 1 and Figure 2.

Theorem 2.2.The graph is a skolem difference mean graph.

Proof. Let n - 2 be the vertices of (n - 2) K2 respectively.

For each vertex label f, the induced edge label f* is calculated as follows:

Therefore, f is a skolem difference mean labeling of Fn ∪ (n - 2) K2 and hence Fn ∪ (n - 2) K2 (n ˃ 2) is a skolem difference mean graph.

A skolem difference mean labeling of F7 ∪ 5K2 is shown in Figure 3.

Theorem 2.3. The graph is a skolem difference mean graph.

For each vertex label f, the induced edge label f* is calculated as follows:

Therefore, f is a skolem difference mean labeling of 3) K2 and hence is a skolem difference mean graph.

A skolem difference mean labeling of is shown in Figure 4.

Theorem 2.4. The graph is a skolem difference mean graph.

Proof. Let n - 1 ) be the vertices of n - 1 ) K2 respectively.

Case(i). n is odd.

For each vertex label f, the induced edge label f* is calculated as follows:

Therefore, and hence f is a skolem difference mean labeling.

Case(ii). n is even.

For each vertex label f, the induced edge label f* is calculated as follows:

Therefore, f is a skolem difference mean labeling and hence Wn (n - 1) K2 is a skolem difference mean graph.

A skolem difference mean labeling of W8 ∪ 7 K2 is shown in Figure 5.

References

[1] F. Harary, Graph theory, Addison Wesley, Massachusetts, (1972). [ Links ]

[2] Joseph A. Gallian, A Dynamic Survey of Graph Labeling, The Electronic Journal of Combinatorics, (2015), \#DS6. [ Links ]

[3] K. Murugan, A. Subramanaian, Labeling of finite union of paths, International Journal of Mathematics and Soft Computing, 2, (1), pp. 99-108, (2012). [ Links ]

[4] K. Murugan, A. Subramanaian, Skolem difference mean labeling of H-graphs, International Journal of Mathematics and Soft Computing, 1, (1), pp. 115-129, (2011). [ Links ]

[5] K. Murugan, Some Results on Skolem Difference Mean Graphs, International Journal of Mathematics and its Applications, 3, (3-D), pp. 75-80, (2015). [ Links ]

[6] D. Ramya, M. Selvi and R. Kalaiyarasi, On skolem difference mean labeling of graphs, International Journal of Mathematical Archive, 4, (12), pp. 73-79, (2013). [ Links ]

[7] M. Selvi, D. Ramya and P. Jeyanthi, Skolem difference mean Graphs, Proyecciones Journal of Mathematics, 34 (3), pp. 243-254, (2015). [ Links ]

[8] S. Somasundaram and R. Ponraj, Mean labelings of graphs, National Academy Science Letter, (26), pp. 210-213, (2003). [ Links ]

Received: January 2017; Accepted: March 2017