Super magic strength of a graph

By G(p, q) we denote a graph having p vertices and q edges, by V and E the vertex set and edge set of G respectively. A graph G(p, q) is said to have an edge magic labeling (valuation) with the constant (magic number) c(f) if there exists a one-to-one and onto function f: V ∪ E → {1, 2, …., p + q} such that f(u)+f(v)+f(uv) = c(f) for all uv ∈ E. An edge magic labeling f of G is called a super magic labeling if f(E) ={1, 2, …., q}. In this paper the concepts of the super magic and super magic strength of a graph are introduced. The super magic strength (sms) of a graph G is defined as the minimum of all constants c′(f) where the minimum is taken over all super magic labeling of G and is denoted by sms(G). This minimum is defined only if the graph has at least one such super magic labeling. In this paper, the super magic strength of some well known graphs P _2n , P _2n+1, K _1,n , B _n,n , < K _1,n : 2 >, P _n ² and (2 _{n + 1})P ₂, C _n and W _n are obtained, where P _n is a path on n vertices, K _1,n is a star graph on n+1 vertices, n-bistar B _n,n is the graph obtained from two copies of K _1,n by joining the centres of two copies of K _1,n by an edge e, if e is subdivided then B _n,n becomes < K _1,n : 2 >, (2 _{n + 1}) P ₂ is 2 _{n + 1} disjoint copies of P ₂, P _n ² is a square graph of P _n . C _n is a cycle on n vertices and W _n = C _n + K ₁ is wheel on n + 1 vertices.