The topological index is usually an invariant derived from the two-dimensional structural graphs to represent the connection of atoms, and its value is independent of the atom number in the molecule. Topological indices are mainly used in studies on structure-activity/property relationships (QSAR/QSPR) [1], that is, to develop models to predict properties/activity of compounds [2,3], such as toxicity [4], density [5,6], specific optical rotations [7,8], enthalpy of formation [9,10], bond dissociation energy [[11], [12], [13]]. Such studies require a correlation between topological indices and properties/activities. In addition, another major application of topological indices is the administration of chemical information [14], which is usually related to the unique identification of molecules, namely, each compound produces a unique topological index value. The corresponding index is called a highly selective molecular topological index. Highly selective molecular topological indices must have high discriminatory power, and although several indices can be used in combination for this purpose, uniqueness tests often only evaluate the discriminatory power of a single topological index. The two applications are not necessarily compatible, as their emphases are different. For example, the Wiener index [15] was successfully used for the prediction of boiling points of the paraffin, but the uniqueness is not sufficient. Herein, the uniqueness of topological indices would be investigated, but the relationship between indices and properties is not our target.

In the beginning, the recognition ability of highly selective molecular topological indices was essentially limited to alkanes containing not more than 20 carbon atoms [16,17], such as the ID index proposed by Randic in 1984 [18]. In 1996, the EAID (Extended Adjacency matrix IDentification) index proposed by Xu L et al. made a significant improvement to distinguish more than two million isomers of C22H46 [19]. In 2015, one of the authors (Zhang Q), in collaboration with Xu L, suggested a highly selective topological index 2-EAID index based on the EAID index, which successfully distinguished more than 14 million isomers of C24H50 [20]. And then the 3-EAID index was suggested in our laboratory in 2016 further distinguished more than 30 million isomers of C25H52 [21]. The 3-EAID index also distinguishes more than 20 million molecules from the ZINC database. The 2-EAID index and the 3-EAID index are both highly distinguishable molecular topological indices. However, the EAID, 2-EAID, and 3-EAID indices are graph theoretical indices; that is, if a molecule can be represented by different chemical graphs mostly due to the expression format of the conjugation, each particular molecular structural graph of the molecule can produce a different value for a graph theoretical index. To solve this problem, one of the authors proposed a highly selective molecular topology index, the ATID index [22]. The ATID index is used to describe the whole molecule and can identify the equivalent molecules but is drawn in the different chemical graphs, such as structural graphs of pyrocatechol in Fig. 1.

Because the positions of double bonds and single bonds in benzene are different, or the benzene is represented in aromatic form, pyrocatechol can be represented by three chemical graphs in Fig. 1. For these three structures, their values are different for a graph theoretical index such as the EAID index, but their values are the same for the ATID index. In addition, to describe the uniqueness of chemical bonds, a highly selective topological index of chemical bonds - bATID index was also suggested by one of the authors. The index was tested using a virtual data set of over 4 million chemical bonds derived from alkanes and a real data set of 3.9 million bonds derived from NCI databases respectively, and no degenerates. The results indicate that the bATID index has a strong discriminant ability of chemical bonds [23].

To describe non-hydrogen atoms in a molecule, a highly selective atomic topological index - aEAID (with the improved d-aEAID index) has been suggested by one of the authors. The uniqueness of the aEAID and d-aEAID indices has also been tested using over 3.8 million atoms derived from alkanes containing 2 to 19 carbon atoms and over 3.8 million atoms derived from the NCI database [24]. The same as the indices of the EAID series, the two atomic indices are graph theoretical indices. To solve this problem, in this research highly selective atomic topological indices-aATID and 2-aATID based on ATID were suggested. The large data set was used to test the uniqueness of the two suggested indices, and the results indicate that both of them possess a high discriminating ability. In addition, the aATID index was applied to the data preprocesses of studies on QSPR for the identification of equivalent atoms. Both series of EAIDs and ATIDs are used to distinguish constitutional isomers, that is, stereoisomers are not considered.