Super magic strength of a graph

Abstract

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 P2n, P2n+1, K1,n, Bn,n, , Pn2 and (2n + 1)P2, Cn and Wn are obtained, where Pn is a path on n vertices, K1,n is a star graph on n+1 vertices, n-bistar Bn,n is the graph obtained from two copies of K1,n by joining the centres of two copies of K1,n by an edge e, if e is subdivided then Bn,n becomes , (2n + 1) P2 is 2n + 1 disjoint copies of P2, Pn2 is a square graph of Pn. Cn is a cycle on n vertices and Wn = Cn + K1 is wheel on n + 1 vertices.