Molecular graph
Encyclopedia
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 to chemical bond
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
vertices deleted.
Molecular graphs can distinguish between structural isomers, compounds which have the same molecular formula but non-isomorphic
graphs - such as isopentane
and neopentane
. On the other hand, the molecular graph normally does not contain any information about the three-dimensional arrangement of the bonds, and therefore cannot distinguish between conformational isomers (such as cis and trans 2-butene
) or stereoisomers (such as D- and L-glyceraldehyde
).
In some important cases (topological index
calculation etc.) the following classical definition is sufficient: molecular graph is connected undirected graph one-to-one corresponded to structural formula of chemical compound so that 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 graphs, in particular, crystals as periodic graphs.
was probably the first to publish results that consider molecular graphs as early as in 1874, even before the introduction of the term "graph
". For the purposes of enumeration of isomer
s, Cayley considered "diagrams" made of ponts 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, further one continues to delete atoms connected by a single link, one arrives to 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" as early as to mid-18th century. In early 18th century, Isaac Newton
's notion of gravity had led to speculative ideas that atoms are held together by some kind of "gravitational force". In particular, since 1758 Scottish chemist William Cullen
in his lectures used what he called "affinity dagrams" 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.
Chemical graph theory
Chemical graph theory is the topology branch of mathematical chemistry which applies graph theory to mathematical modelling of chemical phenomena....
and in 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 molecular graph or chemical graph is a representation of the structural formula
Structural formula
The structural formula of a chemical compound is a graphical representation of the molecular structure, showing how the atoms are arranged. The chemical bonding within the molecule is also shown, either explicitly or implicitly...
of a chemical compound
Chemical compound
A chemical compound is a pure chemical substance consisting of two or more different chemical elements that can be separated into simpler substances by chemical reactions. Chemical compounds have a unique and defined chemical structure; they consist of a fixed ratio of atoms that are held together...
in terms of graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
. A chemical graph is a labeled graph whose vertices correspond to the atoms of the compound and edges correspond to chemical bond
Chemical bond
A chemical bond is an attraction between atoms that allows the formation of chemical substances that contain two or more atoms. The bond is caused by the electromagnetic force attraction between opposite charges, either between electrons and nuclei, or as the result of a dipole attraction...
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
Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...
vertices deleted.
Molecular graphs can distinguish between structural isomers, compounds which have the same molecular formula but non-isomorphic
Graph isomorphism
In graph theory, an isomorphism of graphs G and H is a bijection between the vertex sets of G and H f \colon V \to V \,\!such that any two vertices u and v of G are adjacent in G if and only if ƒ and ƒ are adjacent in H...
graphs - such as isopentane
Isopentane
Isopentane, C5H12, also called methylbutane or 2-methylbutane, is a branched-chain alkane with five carbon atoms. Isopentane is an extremely volatile and extremely flammable liquid at room temperature and pressure. The normal boiling point is just a few degrees above room temperature and...
and neopentane
Neopentane
Neopentane, also called dimethylpropane, is a double-branched-chain alkane with five carbon atoms. Neopentane is an extremely flammable gas at room temperature and pressure which can condense into a highly volatile liquid on a cold day, in an ice bath, or when compressed to a higher...
. On the other hand, the molecular graph normally does not contain any information about the three-dimensional arrangement of the bonds, and therefore cannot distinguish between conformational isomers (such as cis and trans 2-butene
2-Butene
2-Butene is an acyclic alkene with four carbon atoms. It is the simplest alkene exhibiting cis/trans-isomerism ; that is, it exists as two geometrical isomers cis-2-butene , shown at the right, and trans-2-butene , not shown.It is a petrochemical, produced by the catalytic cracking of crude oil...
) or stereoisomers (such as D- and L-glyceraldehyde
Glyceraldehyde
Glyceraldehyde is a triose monosaccharide with chemical formula C3H6O3. It is the simplest of all common aldoses. It is a sweet, colorless, crystalline solid that is an intermediate compound in carbohydrate metabolism...
).
In some important cases (topological index
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 compound. Topological indices are numerical parameters...
calculation etc.) the following classical definition is sufficient: molecular graph is connected undirected graph one-to-one corresponded to structural formula of chemical compound so that 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 graphs, in particular, crystals as periodic graphs.
History
Arthur CayleyArthur Cayley
Arthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....
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 (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...
". For the purposes of enumeration of isomer
Isomer
In chemistry, isomers are compounds with the same molecular formula but different structural formulas. Isomers do not necessarily share similar properties, unless they also have the same functional groups. There are many different classes of isomers, like stereoisomers, enantiomers, geometrical...
s, Cayley considered "diagrams" made of ponts 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, further one continues to delete atoms connected by a single link, one arrives to 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" as early as to mid-18th century. In early 18th century, Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
's notion of gravity 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
William Cullen FRS FRSE FRCPE FPSG was a Scottish physician, chemist and agriculturalist, and one of the most important professors at the Edinburgh Medical School, during its heyday as the leading center of medical education in the English-speaking world.Cullen was also a central figure in the...
in his lectures used what he called "affinity dagrams" to represent forces supposedly existing between pairs of molecules in a chemical reaction. In a 1789 book by William Higgins
William Higgins (chemist)
William Higgins , an Irish chemist, was one of the early proponents of atomic theory. Known mainly for his speculative ideas on chemical combination, William Higgins is popular for the insights his life offers into the emergence of chemistry as a career during the British industrial revolution...
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.