In the fields of chemical graph theory

Chemical graph theory

Chemical graph theory is the topology branch of mathematical chemistry which applies graph theory to mathematical modelling of chemical phenomena....

, molecular topology

Topology (chemistry)

In chemistry, topology provides a convenient way of describing and predicting the molecular structure within the constraints of three-dimensional space. Given the determinants of chemical bonding and the chemical properties of the atoms, topology provides a model for explaining how the atoms...

, and mathematical chemistry

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...

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

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 numbers, allowing some mathematical treatment of the...

 that is calculated based on the molecular graph

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 correspond to the atoms of the compound and edges correspond...

 of a chemical compound. Topological indices are numerical parameters of a graph

Graph (mathematics)

In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...

 which characterize its topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

 and are usually graph invariant. Topological indices are used for example in the development of quantitative structure-activity relationship

Quantitative structure-activity relationship

Quantitative structure–activity relationship or QSPR is the process by which chemical structure is quantitatively correlated with a well defined process, such as biological activity or chemical reactivity.For example, biological activity can be expressed quantitatively as the concentration of a...

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

Correlation

In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....

 with their chemical structure

Chemical structure

A chemical structure includes molecular geometry, electronic structure and crystal structure of molecules. Molecular geometry refers to the spatial arrangement of atoms in a molecule and the chemical bonds that hold the atoms together. Molecular geometry can range from the very simple, such as...

.

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 graph

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 correspond to the atoms of the compound and edges correspond...

s only. More sophisticated topological indices also take into account the hybridization state

Orbital hybridisation

In chemistry, hybridisation is the concept of mixing atomic orbitals to form new hybrid orbitals suitable for the qualitative description of atomic bonding properties. Hybridised orbitals are very useful in the explanation of the shape of molecular orbitals for molecules. It is an integral part...

 of each of the atoms contained in the molecule. Hundreds of indices were introduced. The Hosoya index

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 the number of non-empty matchings plus...

 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

Wiener index

In chemical graph theory, the Wiener index is a topological index of a molecule, defined as the sum of the numbers of edges in the shortest paths in a chemical graph between all pairs of non-hydrogen atoms in a molecule. It was introduced by Harry Wiener in 1947. Wiener index may be calculated...

, 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

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 the number of non-empty matchings plus...

 and Wiener index

Wiener index

In chemical graph theory, the Wiener index is a topological index of a molecule, defined as the sum of the numbers of edges in the shortest paths in a chemical graph between all pairs of non-hydrogen atoms in a molecule. It was introduced by Harry Wiener in 1947. Wiener index may be calculated...

 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

Computational complexity theory

Computational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...

 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
  Wiener index
  In chemical graph theory, the Wiener index is a topological index of a molecule, defined as the sum of the numbers of edges in the shortest paths in a chemical graph between all pairs of non-hydrogen atoms in a molecule. It was introduced by Harry Wiener in 1947. Wiener index may be calculated...
  
  
• Hosoya index
  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 the number of non-empty matchings plus...
  
  
• Estrada index
  Estrada index
  In chemical graph theory, the Estrada index is a topological index of protein folding. The index was first defined by Ernesto Estrada as a measure of the degree of folding of a protein, which is represented as a path-graph weighted by the dihedral or torsional angles of the protein backbone...
  
  
• Randić index

QSAR

QSARs represent predictive model

Statistical model

A statistical model is a formalization of relationships between variables in the form of mathematical equations. A statistical model describes how one or more random variables are related to one or more random variables. The model is statistical as the variables are not deterministically but...

s derived from application of statistical tools correlating biological activity

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 pharmacophore but can be modified by the other...

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

Molecular geometry

Molecular geometry or molecular structure is the three-dimensional arrangement of the atoms that constitute a molecule. It determines several properties of a substance including its reactivity, polarity, phase of matter, color, magnetism, and biological activity.- Molecular geometry determination...

 and/or properties

Molecular property

Molecular properties include the chemical properties, physical properties, and structural properties of molecules, including drugs. Molecular properties typically do not include pharmacological or biological properties of a chemical compound.- See also :...

. QSARs are being applied in many disciplines for example risk assessment

Risk assessment

Risk assessment is a step in a risk management procedure. Risk assessment is the determination of quantitative or qualitative value of risk related to a concrete situation and a recognized threat...

, toxicity prediction, and regulatory decisions in addition to drug discovery

Drug discovery

In the fields of medicine, biotechnology and pharmacology, drug discovery is the process by which drugs are discovered or designed.In the past most drugs have been discovered either by identifying the active ingredient from traditional remedies or by serendipitous discovery...

 and lead optimization

Drug development

Drug development is a blanket term used to define the process of bringing a new drug to the market once a lead compound has been identified through the process of drug discovery...

.

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