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 Hosoy ...

molecular topology A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...

, 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 co ...

, a topological index, also known as a connectivity index, is a type of a

molecular descriptor Molecular descriptors play a fundamental role in chemistry, pharmaceutical sciences, environmental protection policy, and health researches, as well as in quality control, being the way molecules, thought of as real bodies, are transformed into num ...

that is calculated based on the

molecular graph In chemical graph theory and in mathematical chemistry, a molecular graph or chemical graph is a representation of the structural formula of a chemical compound in terms of graph theory. A chemical graph is a labeled graph whose vertices corresp ...

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 discre ...

which characterize its

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

and are usually

graph invariant 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 discr ...

. 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 statistics ...

with their

chemical structure A chemical structure determination includes a chemist's specifying the molecular geometry and, when feasible and necessary, the electronic structure of the target molecule or other solid. Molecular geometry refers to the spatial arrangement of at ...

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

s only. More sophisticated topological indices also take into account the hybridization state of each of the atoms contained in the molecule. The

Hosoya index The Hosoya index, also known as the Z index, of a graph is the total number of matchings in it. The Hosoya index is always at least one, because the empty set of edges is counted as a matching for this purpose. Equivalently, the Hosoya index is ...

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 In chemical graph theory, the Wiener index (also Wiener number) introduced by Harry Wiener, is a topological index of a molecule, defined as the sum of the lengths of the shortest paths between all pairs of vertices in the chemical graph represen ...

, 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

and

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

Hyper-Wiener index In chemical graph theory, the hyper-Wiener index or hyper-Wiener number is a topological index of a molecule, used in biochemistry. The hyper-Wiener index is a generalization introduced by Milan Randić of the concept of the Wiener index, introduc ...

* Estrada index * Randić index
Zagreb indics
* Szeged index *

Padmakar–Ivan index In chemical graph theory, the Padmakar–Ivan (PI) index is a topological index of a molecule, used in biochemistry. The Padmakar–Ivan index is a generalization introduced by Padmakar V. Khadikar and Iván Gutman of the concept of the Wiener ind ...

Gutman index Gutman is a surname. Notable people with the surname include: * Alberto Gutman (born 1959), American politician * Albin Gutman (born 1947), Slovene general * Alexander Gutman (born 1945), Russian film director * Amy Gutman (born 1960), American n ...

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 ''modulus'', a measure. Models c ...

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 deter ...

and/or

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

. QSARs are being applied in many disciplines for example

risk assessment Broadly speaking, a risk assessment is the combined effort of: # identifying and analyzing potential (future) events that may negatively impact individuals, assets, and/or the environment (i.e. hazard analysis); and # making judgments "on the to ...

, 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 by ...

and

lead optimization Hit to lead (H2L) also known as lead generation is a stage in early drug discovery where small molecule hits from a high throughput screen (HTS) are evaluated and undergo limited optimization to identify promising lead compounds. These lead compo ...

. 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