Strongly Multiplicative Labeling in the Context of Arbitrary Supersubdivision

We investigate some new results for strongly multiplicative labeling of graph. We prove that the graph obtained by arbitrary supersubdivision of tree T , grid graph Pn × Pm, complete bipartite graph Km,n, Cn Pm and one-point union of m cycle of length n are strongly multiplicative.


Introduction
We begin with simple, finite, undirected and connected graph G = (V, E).In the present work T , P n × P m and K m,n denote the tree, grid graph, and complete bipartite graph respectively.C n P m is the graph obtained by identifying an end point of P m with every vertex of cycle C n .One point union of m cycles of length n denoted as C (m)  n is the graph obtained by identifying one vertex of each cycles.If V 1 and V 2 are two partitions correspond to complete bipartite graph (u, v).For all other terminology and notations we refer to (Harary, F., 1972).We will give brief summary of definitions and other information which are useful for the present investigations.
Definition 1.1 Let G be a graph with q edges.A graph H is called a supersubdivision of G if H is obtained from G by replacing every edge e i of G by a complete bipartite graph K 2,m i for some m i , 1 ≤ i ≤ q in such a way that the end vertices of each e i are merged with the two vertices of 2-vertices part of K 2,m i after removing the edge e i from graph G.
A supersubdivision H of G is said to be an arbitrary supersubdivision of G if every edge of G is replaced by an arbitrary K 2,m (m may vary for each edge arbitrarily).Arbitrary supersubdivision of G is denoted by S S (G).
Definition 1.2 If the vertices of the graph are assigned values subject to certain conditions then it is known as graph labeling.
Most interesting graph labeling problems have following three important characteristics.
1. a set of numbers from which the labels are chosen; 2. a rule that assigns a value to each edges; Labeled graph have variety of applications in coding theory, particularly for missile guidance codes, design of good radar type codes and convolution codes with optimal autocorrelation properties.Labeled graph plays vital role in the study of Xray crystallography, communication network and to determine optimal circuit layouts.A systematic study on applications of graph labeling is reported in (Bloom, G., 1977, p. 562-570).
Definition 1.3 A graph G = (V, E) with p vertices is said to be multiplicative if the vertices of G can be labeled with p distinct positive integers such that label induced on the edges by the product of labels of end vertices are all distinct.
Multiplicative labeling was introduced in (Beineke, L., 2001, p.63-75) where it is shown that every graph G admits multiplicative labeling and strongly multiplicative labeling is defined as follows.
Definition 1.4 A graph G = (V, E) with p vertices is said to be strongly multiplicative if the vertices of G can be labeled with p distinct integers 1, 2, ...p such that label induced on the edges by the product of labels of the end vertices are all distinct.
In the present investigations we prove that the graphs obtained by arbitrary supersubdivision of tree T , grid graph P n × P m , complete bipartite graph K m,n , C n P m and C (m)  n are strongly multiplicative for all n and m.

Main Results
Theorem-2.1:Arbitrary supersubdivisions of tree T are strongly multiplicative.
Proof: Let T be the tree with n vertices.Arbitrary supersubdivision SS(T ) of tree T obtained by replacing every edge of tree with K 2,m i and we denote such graph by G.
Denote the vertex with minimum eccentricity as v 1 .Then v 2 will be the vertex which is at 1-distance apart from v 1 .If there are more than one such vertices then throughout the work we will follow one of the direction ( clockwise or anticlockwise) and denote them as v 3 , v 4 , . . . .Next consider the vertices which are at 2-distance apart from v 1 , 3-distance apart from v 1 and so on.(e.g. if there are seven vertices and two vertices are at distance 1-apart, one vertex is at distance 2apart and three vertices are at distance 3-apart respectively form v 1 .In this situation the vertices which are at 1-distance apart from v 1 will be identified as v 2 and v 3 , the vertex which is at distance 2-apart will be identified as v 4 and the vertices which are at distance 3-apart will be identified as v 5 , v 6 and v 7 .)We define vertex labeling f : V(G) → {1, 2 . . .K + n} as follows.
For any 1 Then the graph G under consideration admits strongly multiplicative labeling.
Illustration 2.2: In Fig. 2 strongly multiplicative labeling of SS(T ) corresponding to tree T of Fig. 1 is shown where n = 13 and K = 26.
Theorem 2.3: Arbitrary supersubdivisions of complete bipartite graph K m,n are strongly multiplicative.Proof: Let v 1 , v 2 , v 3 , . . .v m be the vertices of m-vertices part and v m+1 , v m+2 , v m+3 , . . .v m+n be the vertices of n-vertices part of K m,n .Arbitrary supersubdivision SS(K m,n ) of K m,n obtained by replacing every edge of K m,n with K 2,m i and we denote such graph by G. Let K = m i (1 ≤ i ≤ mn).Let u j be the vertices which are used for arbitrary supersubdivision, where 1 ≤ j ≤ K.We denote vertices by u j which are used for supersubdivision of edges v 1 v m+1 , v 1 v m+2 , . . .v 1 v m+n , v 2 v m+1 , . . .v n v m+n .Let p o be the highest prime less than K + m + n.We define vertex labeling f : V(G) → {1, 2 . . .K + m + n} as follows. If