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In 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 Hoso ...
and in mathematical chemistry, a molecular graph or chemical graph is a representation of the structural formula
The structural formula of a chemical compound is a graphic representation of the molecular structure (determined by structural chemistry methods), showing how the atoms are possibly arranged in the real three-dimensional space. The chemical bond ...
of a chemical compound
A chemical compound is a chemical substance composed of many identical molecules (or molecular entities) containing atoms from more than one chemical element held together by chemical bonds. A molecule consisting of atoms of only one element ...
in terms of graph theory
In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
. A chemical graph is a labeled graph
In the mathematical discipline of graph theory, a graph labelling is the assignment of labels, traditionally represented by integers, to edges and/or vertices of a graph.
Formally, given a graph , a vertex labelling is a function of to a set ...
whose vertices correspond to the atoms of the compound and edges correspond to chemical bond
A chemical bond is a lasting attraction between atoms or ions that enables the formation of molecules and crystals. The bond may result from the electrostatic force between oppositely charged ions as in ionic bonds, or through the sharing of ...
s. Its vertices are labeled with the kinds of the corresponding atoms and edges are labeled with the types of bonds. For particular purposes any of the labelings may be ignored.
A hydrogen-depleted molecular graph or hydrogen-suppressed molecular graph is the molecular graph with hydrogen
Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-to ...
vertices deleted.
In some important cases (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 co ...
calculation etc.) the following classical definition is sufficient: a molecular graph is a connected, undirected graph which admits a one-to-one correspondence with the structural formula of a chemical compound in which the vertices of the graph correspond to atoms of the molecule and edges of the graph correspond to chemical bonds between these atoms. One variant is to represent materials as infinite Euclidean graph
Geometric graph theory in the broader sense is a large and amorphous subfield of graph theory, concerned with graphs defined by geometric means. In a stricter sense, geometric graph theory studies combinatorial and geometric properties of geometr ...
s, in particular, crystals as periodic graphs.
History
Arthur Cayley
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics.
As a child, Cayley enjoyed solving complex maths problem ...
was probably the first to publish results that consider molecular graphs as early as in 1874, even before the introduction of the term "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 ...
". For the purposes of enumeration of isomer
In chemistry, isomers are molecules or polyatomic ions with identical molecular formulae – that is, same number of atoms of each element – but distinct arrangements of atoms in space. Isomerism is existence or possibility of isomers.
Is ...
s, Cayley considered "diagrams" made of points labelled by atoms and connected by links into an assemblage. He further introduced the terms plerogram and kenogram, which are the molecular graph and the hydrogen-suppressed molecular graph respectively. If one continues to delete atoms connected by a single link further, one arrives at a mere kenogram, possibly empty.
Danail Bonchev in his ''Chemical Graph Theory'' traces the origins of representation of chemical forces by diagrams which may be called "chemical graphs" to as early as the mid-18th century. In the early 18th century, Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
's notion of gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
had led to speculative ideas that atoms are held together by some kind of "gravitational force". In particular, since 1758 Scottish chemist William Cullen
William Cullen FRS FRSE FRCPE FPSG (; 15 April 17105 February 1790) was a Scottish physician, chemist and agriculturalist, and professor at the Edinburgh Medical School. Cullen was a central figure in the Scottish Enlightenment: He was ...
in his lectures used what he called "affinity diagrams" to represent forces supposedly existing between pairs of molecules in a chemical reaction. In a 1789 book by William Higgins similar diagrams were used to represent forces within molecules. These and some other contemporary diagrams had no relation to chemical bonds: the latter notion was introduced only in the following century. Danail Bonchev (1991) "Chemical Graph Theory: Introduction and Fundamentals"
See also
*Chemical graph generator
A chemical graph generator is a software package to generate computer representations of chemical structures adhering to certain boundary conditions. The development of such software packages is a research topic of cheminformatics. Chemical graph g ...
References