In a recent paper [Vukičević et al., J. Math. Chem. {\bf 48} (2010) 395-�400] a novel molecular-graph-based structure descriptor, named one-two descriptor ($OT$), was introduced. $OT$ is the sum of vertex contributions, such that each pendent vertex contributes 1, each vertex of degree two adjacent to a pendent vertex contributes 2, and each vertex of degree higher than two also contributes 2. Vertices of degree two, not adjacent to a pendent vertex, do not contribute to $OT$. Vukučević et al. established lower and upper bounds on $OT$ for trees. We now give lower and upper bounds on $OT$ for general graphs, and also characterize the extremal graphs. The bounds of Vukičević et al. for trees follows as a special case. Moreover, we give another upper bound on $OT$ for trees.