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 Szeged index is a topological index 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 Szeged index, introduced by

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 ...

, generalizes 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 Harry Wiener (Oct. 29, 1924 in Vienna, Austria – Nov. 8, 1998 in New York City, USA) was an Austrian-American chemist, physician and psychologist, a pioneer in cheminformatics and chemical graph theory, and a long-time employee at Pfizer. Edu ...

. The Szeged index of a connected graph ''G'' is defined as :

Sz(G)=\sum_ n_1(e\mid G)n_2(e\mid G),

If ''e'' is an edge of ''G'' connecting vertices ''u'' and ''v'', then we write ''e'' = ''uv'' or ''e'' = ''vu''. For

e=uv\in E(G)

, let

n_1(e\mid G)

and

n_2(e\mid G)

be respectively the number of vertices of ''G'' lying closer to vertex ''u'' than to vertex ''v'' and the number of vertices of ''G'' lying closer to vertex ''v'' than to vertex ''u''. Szeged index plays an important role in

information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist a ...

. One way to measure a network structure is through the so-called topological indices. Szeged index has been shown to correlate well with numerous

biological 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 in ...

and physicochemical properties.

Examples

The Szeged index of Dendrimer Nanostar of the following figure can be calculated by. :

Sz(T_n) = 1620n\cdot4^n - 2376\cdot4^n + 2862\cdot2^n-432, \quad n \geq 0.

References

{{reflist, colwidth=30em Mathematical chemistry Cheminformatics Graph invariants