The topological bond order is a bond-order-like quantity, put forward in the 1970s. It is defined as p(rs)(T) = Z(G(rs))/Z(G), where G is the molecular graph, G(rs) 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 p(rs)(T) is known, its properties were studied only to a limited degree. We now introduce a modified topological bond order, p(rs)(T), that can (easily) be calculated from the eigenvalues of G and G(rs). For acyclic systems, p(rs)(T) = p(rs)(T). In the case of polycyclic systems a reasonably accurate linear correlation exists between p(rs)(T) and p(rs)(T). Thus, by studying p(rs)(T) the main properties of p(rs)(T) can be established. © 2005 Elsevier B.V. All rights reserved.