Let G = (V, E), V = {v1, v2, . . . , vn}, be a simple graph of order n and size m, with vertex–degree sequence ∆ = d1 ≥ d2 ≥ · · · ≥ dn = δ > 0. and let s1 ≥ s2 ≥ · · · ≥ sn, si = di − δ + 1 and c1 ≤ c2 ≤ · · · ≤ cn, ci = ∆ − di + 1, be two vertex–degree like sequences. By analogy with the inverse degree graph invariant, ID(G) = ∑n i=1 1 di , the delta inverse degree and reverse inverse degree indices are defined, respectively, as δID(G) = ∑n i=1 1 si and RID(G) = ∑n i=1 1 ci . In this paper we determine sharp bounds on δID(G) and RID(G) and the extremal graphs are characterized.