On the Basis of Invariants of Labeled Molecular Graphs

It is proved that any molecular graph invariant (that is any topological index) can be uniquely represented as (1) a linear combination of occurrence numbers of some substructures (fragments), both connected and disconnected, or (2) a polynomial on occurrence numbers of connected substructures of corresponding molecular graph. Besides, any (0,l)-valued molecular graph invariant can be uniquely represented as a linear combination (in the terms of logic operations) of some basic (0, 1)-valued invariants indicating the presence of some substructures in the chemical structure. Thus, the occurrence numbers of substructures in a structure (or numbers indicating the presence or absence of substructures in a structure for the case of (0,l)-valued invariants) are shown to constitute the basis of invariants of labeled molecular graphs. A possibility to use these results for the mathematical justification of substructures-based methods in the “structure-property” problem is also discussed.