Bounds on Sum Number in Graphs Hiroshi

A simple undirected graph G is called a sum graph if there is a labeling L of the vertices of G into distinct positive integers such that any two vertices u and v of G are adjacent if and only if there is a vertex w with label L(w) = L(u) + L(v). The sum number (H) of a graph H = (V; E) is the least integer r such that graph G consisting of H and r isolated vertices is a sum graph. It is clear that (H) jEj. In this paper, we discuss general upper and lower bounds on the sum number. In particular, we prove that the average of (H) over all graphs H = (V; E) with xed jV j and jEj is at least jEj 0 3jV j log jV j log((jV j 2)=jEj) 0 jV j 0 1. In other words, for most graphs, (H) = (jEj). 1 1 Introduction The notion of a sum graph was rst introduced by Harray [6]. From a practical point of view, sum graph labeling can be used as the compressed representation of a graph, a data structure for representing the graph. The data compression is important not only for saving memory space but also for speeding up some graph algorithm when adapted to work with the compressed representation of the input graph (for example, see [4]). There have been several papers determining or bounding the sum number of particular classes of graphs H = (V; E) (n = jV j, m = jEj): (K n) = 2n 0 3 for complete graphs K n (n 4) [1] (K p;q) = d