GTI-simplex is a new methodology that combines the generalized topological indices and the down hill simplex optimization procedure to search for optimized quantitative structure-property relationship models (Chem. Phys. Lett. 2005, 410, 343). In this study, the fundamental role of the graph topological distance inducing a local shell structure on vertexes and a detailed derivation of the GTI-decomposition in terms of the so-called "geodesic-brackets", i.e., functions that mix the local shell structure for different vertexes are presented. Applications of the GTI-simplex to a set of physicochemical properties covering those depending on intramolecular and/or intermolecular interactions are included. GTI-simplex has showed to be a very effective methodology for the description of different properties from a unified point of view. No ad hoc definition for topological index is required to each property as in the traditional use of topological indices or other molecular descriptors to QSAR/QSPR studies.