Let G = (V, E), V = {1, 2, . . . , n} be a simple connected graph with n vertices, m edges and a sequence of vertex degrees d 1 ≥ d 2 ≥ · · · ≥ d n > 0, d i = d(i). The irregularity measure of graph is defined as irr(G) = ∑ i∼ j d i − d j , where i ∼ j denotes adjacency of vertices i and j. New upper bounds for irr(G) are obtained.

Let G = (V, E), V = {v1, v2,..., vn}, be a simple graph without isolated vertices, with the sequence of vertex degrees d1 ? d2 ?...? dn > 0, di = d(vi). If vertices vi and vj are adjacent in G, we write i ~ j, otherwise we write i ~/ j. The inverse degree topological index of G is defined to be ID(G) = ?ni=1 1/di = ? i~j (1/d2i + 1/d2j), and the inverse degree coindex ?ID(G) = ? i~/j(1/d2i + 1/d2j). We obtain a number of inequalities which determine bounds for the ID(G) and ?ID(G) when G is a tree.