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.
