One-two descriptor of graphs

In a recent paper [Vukičević et al., J. Math. Chem. 48 (2010) 395-400] a novel molecular-graph-based structure descriptor, named one-two descriptor (OT), was introduced. OT is the sum of vertex contri-butions, 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.