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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 ...
, the Padmakar–Ivan (PI) index is a topological index
In the fields of chemical graph theory, molecular topology, and mathematical chemistry, 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 compo ...
of a molecule
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 bioch ...
, used in biochemistry
Biochemistry or biological chemistry is the study of chemical processes within and relating to living organisms. A sub-discipline of both chemistry and biology, biochemistry may be divided into three fields: structural biology, enzymology and ...
. The Padmakar–Ivan index is a generalization introduced by Padmakar V. Khadikar and Iván Gutman
Iván Gutman (born in 1947) is a Serbian chemist and mathematician.
Life and work
Gutman was born in Sombor, Yugoslavia in a Bunjevac family. In 1970 he graduated chemistry from the University of Belgrade where he worked a short time as an assis ...
of the concept of 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 ...
, introduced by Harry Wiener. The Padmakar–Ivan index of a graph ''G'' is the sum over all edges ''uv'' of ''G'' of number of edges which are not equidistant from ''u'' and ''v''.
Let ''G'' be a graph and ''e'' = ''uv'' an edge of ''G''. Here denotes the number of edges lying closer to the vertex ''u'' than the vertex ''v'', and is the number of edges lying closer to the vertex ''v'' than the vertex ''u''. The Padmakar–Ivan index of a graph ''G'' is defined as
:
The PI index is very important in the study of quantitative structure–activity relationship
Quantitative structure–activity relationship models (QSAR models) are regression or classification models used in the chemical and biological sciences and engineering. Like other regression models, QSAR regression models relate a set of "predic ...
for the classification models used in the chemical
A chemical substance is a form of matter having constant chemical composition and characteristic properties. Some references add that chemical substance cannot be separated into its constituent elements by physical separation methods, i.e., wi ...
, biological sciences
Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary ...
, engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
, and nanotechnology
Nanotechnology, also shortened to nanotech, is the use of matter on an atomic, molecular, and supramolecular scale for industrial purposes. The earliest, widespread description of nanotechnology referred to the particular technological goal o ...
.
Examples
The PI index of Dendrimer Nanostar of the following figure can be calculated by. :
References