In the fields of

chemical graph theory Chemical graph theory is the topology branch of mathematical chemistry which applies graph theory to mathematical modelling of chemical phenomena. The pioneers of chemical graph theory are Alexandru Balaban, Ante Graovac, Iván Gutman, Haruo ...

, molecular topology, and

mathematical chemistry Mathematical chemistry is the area of research engaged in novel applications of mathematics to chemistry; it concerns itself principally with the mathematical modeling of chemical phenomena. Mathematical chemistry has also sometimes been called ...

, a topological index, also known as a connectivity index, is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound. Topological indices are numerical parameters of a

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

which characterize its

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

and are usually graph invariant. Topological indices are used for example in the development of

quantitative structure-activity relationship Quantitative may refer to: * Quantitative research, scientific investigation of quantitative properties * Quantitative analysis (disambiguation) * Quantitative verse, a metrical system in poetry * Statistics, also known as quantitative analysis ...

s (QSARs) in which the biological activity or other properties of molecules are

correlated In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistic ...

with their

chemical structure A chemical structure of a molecule is a spatial arrangement of its atoms and their chemical bonds. Its determination includes a chemist's specifying the molecular geometry and, when feasible and necessary, the electronic structure of the target m ...

Calculation

Topological descriptors are derived from hydrogen-suppressed molecular graphs, in which the atoms are represented by vertices and the bonds by edges. The connections between the atoms can be described by various types of topological matrices (e.g., distance or adjacency matrices), which can be mathematically manipulated so as to derive a single number, usually known as graph invariant, graph-theoretical index or topological index. As a result, the topological index can be defined as two-dimensional descriptors that can be easily calculated from the molecular graphs, and do not depend on the way the graph is depicted or labeled and no need of energy minimization of the chemical structure.

Types

The simplest topological indices do not recognize double bonds and atom types (C, N, O etc.) and ignore hydrogen atoms ("hydrogen suppressed") and defined for connected undirected molecular graphs only. More sophisticated topological indices also take into account the hybridization state of each of the atoms contained in the molecule. The Hosoya index is the first topological index recognized in chemical graph theory, and it is often referred to as "the" topological index. Other examples include the Wiener index, Randić's molecular connectivity index, Balaban's J index, and the TAU descriptors. The extended topochemical atom (ETA) indices have been developed based on refinement of TAU descriptors.

Global and local indices

Hosoya index and Wiener index are global (integral) indices to describe entire molecule, Bonchev and Polansky introduced local (differential) index for every atom in a molecule. Another examples of local indices are modifications of Hosoya index.

Discrimination capability and superindices

A topological index may have the same value for a subset of different molecular graphs, i.e. the index is unable to discriminate the graphs from this subset. The discrimination capability is very important characteristic of topological index. To increase the discrimination capability a few topological indices may be combined to superindex.

Computational complexity

Computational complexity In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations ...

is another important characteristic of topological index. The Wiener index, Randic's molecular connectivity index, Balaban's J index may be calculated by fast algorithms, in contrast to Hosoya index and its modifications for which non-exponential algorithms are unknown.

List of topological indices

* Wiener index * Hosoya index * Hyper-Wiener index * Estrada index * Randić index
Zagreb indices
* Szeged index * Padmakar–Ivan index * Gutman index * sombor index * Harmonic index * Arithmetic index * Atom bond connectivity index * Merrifield-Simmons index

Application

QSAR

QSARs represent predictive

model A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , . Models can be divided in ...

s derived from application of statistical tools correlating

biological activity In pharmacology, biological activity or pharmacological activity describes the beneficial or adverse effects of a drug on living matter. When a drug is a complex chemical mixture, this activity is exerted by the substance's active ingredient or ...

(including desirable therapeutic effect and undesirable side effects) of chemicals (drugs/toxicants/environmental pollutants) with descriptors representative of

molecular structure Molecular geometry is the three-dimensional arrangement of the atoms that constitute a molecule. It includes the general shape of the molecule as well as bond lengths, bond angles, torsional angles and any other geometrical parameters that det ...

and/or

properties Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property. Property may also refer to: Philosophy and science * Property (philosophy), in philosophy and logic, an abstraction characterizing an ...

. QSARs are being applied in many disciplines for example

risk assessment Risk assessment is a process for identifying hazards, potential (future) events which may negatively impact on individuals, assets, and/or the environment because of those hazards, their likelihood and consequences, and actions which can mitigate ...

, toxicity prediction, and regulatory decisions in addition to

drug discovery In the fields of medicine, biotechnology, and pharmacology, drug discovery is the process by which new candidate medications are discovered. Historically, drugs were discovered by identifying the active ingredient from traditional remedies or ...

and lead optimization. For example, ETA indices have been applied in the development of predictive QSAR/QSPR/QSTR models.; ; ; ;

References

External links

* Software for calculating various topological indices
''GraphTea''
Theoretical chemistry Mathematical chemistry Graph invariants Cheminformatics