An important area of theoretical drug design research is quantitative structure activity relationship (QSAR)
using structural invariants. The impetus for this research trend comes from various directions. Researchers in chemical
documentation have searched for a set of invariants which will be more convenient than the adjacency matrix (or
connection table) for the storage and comparison of chemical structures . Molecular structure can be looked upon as the
representation of the relationship among its various constituents. The term molecular structure represents a set of nonequivalent
and probably disjoint concepts . There is no reason to believe that when we discuss diverse topics (e.g.
chemical synthesis, reaction rates, spectroscopic transitions, reaction mechanisms, and ab initio calculations) using the
notion of molecular structure, the different meanings we attach to the single term molecular structure originate from the
same fundamental concept. On the contrary, there is a theoretical and philosophical basis for the non-homogeneity of
concepts covered by the term molecular structure.
In the context of molecular science, the various concepts of molecular structure (e.g. classical valence bond
representations, various chemical graph-theoretic representations, ball and spoke model of a molecule, representation of a
molecule by minimum energy conformation, semi symbolic contour map of a molecule, or symbolic representation of
chemical species by Hamiltonian operators) are model objects  derived through different abstractions of the same
chemical reality. In each instance, the equivalence class (concept or model of molecular structure) is generated by
selecting certain aspects while ignoring some unique properties of those actual events. This explains the plurality of the
concept of molecular structure and their autonomous nature, the word autonomous being used in the same sense that one
concept is not logically derived from the other.
At the most fundamental level, the structural model of an assembled entity (e.g. a molecule consisting of atoms) may be
defined as the pattern of relationship among its parts as distinct from the values associated with them .
Constitutional formulae of molecules are graphs where vertices represent the set of atoms and edges represent chemical
bonds . The pattern of connectedness of atoms in a molecule is preserved by constitutional graphs. A graph (more
correctly a non-directed graph) G = [V, E] consists of a finite non-empty set V of points together with a prescribed set E
of unordered pairs of distinct points of V . Thus the mathematical characterization of structures represents structural
invariants having successful applications in chemical documentation, characterization of molecular branching,
enumeration of molecular constitutional associated with a particular empirical formula, calculation of quantum chemical
parameters for the generation of quantitative structure-property-activity correlations . Kier developed a number of
structural invariants which are now-a-days called as topological indices with wide range of practical applications for
QSAR and drug design. The present paper is restricted to the review of Kier-Hall topological indices for QSAR and
anticancer drug design for 2,5-bis(1-aziridinyl) 1,4-benzoquinone (BABQ) , pyridopyrimidine , 4-anilinoquinazoline
 and 2-Phenylindoles  compounds utilizing various statistical multivariate regression analyses.