Some properties of the topological bond order

The topological bond order is a bond-order-like quantity, put forward in the 1970s. It is defined as prsT=Z(Grs)/Z(G), where G is the molecular graph, Grs is obtained from G by deleting from it the adjacent vertices labelled by r and s, and Z stands for the respective topological (Hosoya) index. Because no easy way for the calculation of prsT is known, its properties were studied only to a limited degree. We now introduce a modified topological bond order, p˜rsT, that can (easily) be calculated from the eigenvalues of G and Grs. For acyclic systems, p˜rsT=prsT. In the case of polycyclic systems a reasonably accurate linear correlation exists between p˜rsTandprsT. Thus, by studying p˜rsT the main properties of prsT can be established.