On The Edge Distribution Of A Graph

We investigate a graph function which is related to the local density, the maximal cut and the least eigenvalue of a graph. In particular it enables us to prove the following assertions: Let p 3 be integer, c 2 (0; 1=2) and G be a Kp-free graph on n vertices with e cn2 edges. There exists a positive constant = (c; p) such that a) some bn=2c subset of V (G) induces at most c 4 n edges (this answers a question of Paul Erd1⁄2os); b) G can be made bipartite by the omission of at most c 2 n edges. 1 Notation and conventions We consider only simple …nite, undirected graphs and use the standard graphic terminology and notation of [1]. All graphs are assumed to be de…ned on the vertex set f1; 2; :::ng : By V (G) we denote the set of vertices of