Graph theory based descriptors of molecular structure play important role in QSPR/ QSAR models. This chapter reviews some attempts to optimize the characterization of molecular structure via an integrated representation that accounts in a systemic manner for the contributions of all substructures. In its simplest version this approach counts the subgraphs of all sizes, the resulted single number being shown to be a very sensitive measure of structural complexity. The most complete version builds (i) an ordered set of counts of subgraphs of increasing number of edges, (ii) weights each subgraph with the value of selected graph-invariant, building a weighted ordered set, and (iii) sums up all the subgraph contributions to produce the overall value of the graph-invariant. The invariants tested include vertex degrees, vertex distances, and the graph non-adjacency numbers, the corresponding overall topological indices being called overall connectivity, overall Wiener, overall Zagreb and overall Hosoya indices. Their properties are analyzed in detail in acyclic and cyclic graphs. It is shown that they all are reliable measures of molecular structural complexity, increasing in value with the basic complexifying patterns of branching and cyclicity of molecular skeleton. The structure-property models derived for 10 physicochemical properties of alkane compounds show considerable improvement compared to models derived from molecular connectivity indices. The latest extension of these ideas is demonstrated with extended connectivities, walk counts, and Bourgas indices, the latter of which are the first integrated measures of graph complexity and vertex centrality.
Keywords: Molecular structure, molecular topology, molecular descriptors, graph theory, topological indices, overall connectivity, overall Wiener index, overall Zagreb indices, overall Hosoya index, Bourgas indices, structural complexity, structure-property models, vertex centrality, molecular branching, molecular cyclicity.