Independence in connected graphs

We prove that if G = (V-G E-G) is a finite simple and undirected graph with K components and independence number alpha(G) then there exist a positive integer k is an element of N and a function f V-G -> N-0 with non-negative integer values such that f(u) <= d(G)(u) for u is an element of V-G alpha(G) >= k >= Sigma u is an element of V-G 1/d(G)(u)+1-f(u) and Sigma u is an element of V(G)f(u)>= 2(k - k) This result is a best possible improvement of a result due to Harant and Schiermeyer [J Harant I Schiermeyer On the independence number of a graph in terms of order and size Discrete Math 232 (2001) 131-138] and implies that alpha(G)/n(G) >= 2/(d(G) + 1 + 2/n(G)) + root(d(G) + 1 + 2/n(G))(2) for connected graphs G of order n(G) average degree d(G) and independence number alpha(G) (C) 2010 Elsevier B V All rights reserved