240 on Graphs of Irregularity Strength 2

We consider undirected graphs without loops or multiple edges. A weighting of a graph G is an assignment of a positive integer w( e) to each edge of G. For a vertex x€V(G), the (weighted) degree d(x) is the sum ofweights on the edges ofG incident to x. The irregularity strength s( G) of a graph G was introduced by Chartrand et al. in [1] a.s the minimum integer t such that G has a weighting with the following two properties: (i) w( e) ::; t for all e€E( G) (ii) d(x) =/: d(y) if x, y€V(G), x =/: y Since every graph has two vertices of the same degree, s(G) 2:: 2 for all graphs G. Some results and problems concerning the irregularity strength of graphs appear in [1] arid [2]. Assume that G is a graph with IV (G) I = n and s (c) = 2. We will determine the minimum and maximum number of edges in a graph of irregularity strength 2. We prove that IE(G)I2:: f(n 1)/81 and establish this bound is sharp (Theorem 1). Concerning the upper bound of IE(G)I, Jacobson and Lehel conjectured that .