# 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 to chemical bonds. Its vertices are labeled with the kinds of the corresponding atoms and edges are labeled with the types of bonds.[1] 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 geometric isomers (such as *cis* and *trans* 2-butene) or other 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.[2] One variant is to represent materials as infinite Euclidean graphs, in particular, crystals as periodic graphs.[3]

## History

Arthur Cayley was probably the first to publish results that consider molecular graphs as early as in 1874, even before the introduction of the term "graph".[4] For the purposes of enumeration of isomers, Cayley considered "diagrams" made of points labelled by atoms and connected by links into an assemblage. He further introduced the terms **plerogram** and **kenogram**,[5] 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.[6]

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'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 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.[7]

## References

- IUPAC,
*Compendium of Chemical Terminology*, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "molecular graph". doi:10.1351/goldbook.MT07069 -
*Chemical Applications of Topology and Graph Theory*, ed. by R. B. King, Elsevier, 1983 - Sunada T. (2012),
*Topological Crystallography ---With a View Towards Discrete Geometric Analysis---", Surveys and Tutorials in the Applied Mathematical Sciences, Vol. 6, Springer* - A. Cayley,
*Phil. Mag.*, 1874, 47, 444-446, as quoted in N. L. Biggs, E. K. Lloyd and R. J. Wilson, "Graph Theory 1736-1936",*Clarendon Press*, Oxford, 1976;*Oxford University Press*, 1986, ISBN 0-19-853916-9 - Derived from the Greek words πλήρης,
*pleres*"full" and κενός,*kenos*"empty", respectively. - Biggs, Lloyd, Wilson, p. 61
- Danail Bonchev (1991) "Chemical Graph Theory: Introduction and Fundamentals" ISBN 0-85626-454-7