The notation of a sum graph was introduced by Harary in 1990. A graph is said to be a sum graph if there exists a bijective labeling f from V to a set of positive integers S such that xy E E if and only if f(x) + f(y)E S. This paper defines for any connected graph G the edge function, the edge product function, the edge product graph and the unit edge product graph. The properties of edge product graph are also discussed, and then the concepts are extended to prove the theorems.

Harary (1990) introduced sum graphs. Alon and Scheinerman (1992) generalized sum graphs by replacing the condition f(x) + f(y) E S with g(f(x), f(y)) E S, where g is an arbitrary symmetric polynomial. Ellingham (1993) proved the conjecture of Harary that S(Kn) = (2n – 3). Harary (1994), generalized sum graphs by permitting S to be any set of integers. He calls these graphs integral sum graphs. Chen (1998) conjectures that all trees are integral sum graphs. He proved several properties of sum labeling of the graph with E(G) = |V(G)|– 1. Stewart (1996) studied various ways to label the edges of a graph. Several well-known graceful graphs have been examined by Bony and Murty (1976). For a detailed account on variation of sum graphs one can refer to Gallian (1994 and 2000). The concept of an antimagic edge labeling was introduced by Bodendiek and Walther (1998). This paper gives an idea about edge product graph, the edge analogue of product graph and its properties. A graph is said to be an edge product graph if the edges of G can be labeled with distinct positive integers such that the product of all the labels of the edges incident on a vertex is again an edge label of G, and if the product of any collection of edges is a label of an edge in G, then they are incident on a vertex.

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