LINEAR BOUND ON THE IRREGULARITY STRENGTH AND THE TOTAL VERTEX IRREGULARITY STRENGTH OF GRAPHS

Let G be a simple graph of order n with no isolated edges and at most one isolated vertex. For a positive integer w, a w-weighting of G is a function f : E(G) -> {1, 2, ... , w}. An irregularity strength of G, s(G), is the smallest w such that there is a w-weighting of G for which Sigma(e:u is an element of e) f(e) not equal Sigma(e:u is an element of e) f(e) for all pairs of different vertices u, v is an element of V (G). We prove that s(G) < 112 n/delta + 28, where d is the minimum degree of G. For d-regular graphs, we strengthen this to s(G) < 40 n/d + 11. These upper bounds represent improvements of many existing ones. Similar results concerning the " total" version of the irregularity strength are also discussed.