Abstract
Let G be a simple connected non-trivial graph of order n, size m, and vertex degree sequence (d(1), d(2), ..., d(n)). The first Zagreb index M-1, forgotten index F and inverse degree I D are the graph invariants defined as M-1(G) = Sigma(n)(i=1) d(i)(2), F(G) = Sigma(n)(i=1) d(i)(3) and I D(G) = Sigma(n)(i=1) 1/d(i), respectively. A graph is said to be regular if all of its vertices have the same degree and otherwise, it is called a nonregular graph. For the quantitative topological characterization of the nonregularity of graphs, the graph invariants s(G) = Sigma(n)(i=1) vertical bar d(i) - 2m/n vertical bar and Var(G) = 1/n Sigma(n)(i=1) (d(i) - 2m/n)(2) may be used. In this paper, some upper bounds for s(G) that reveal connections between s(G) and M-1(G), F(G), I D(G), Var(G) are obtained.