Molecular graph

In chemical graph theory, 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 colored graph whose vertices correspond to the atoms of the compound and edges correspond to chemical bonds. Its vertices are colored with the kinds of the corresponding atoms and edges are colored with the types of bonds. For particular purposes any of the colorings may be ignored.

A hydrogen-depleted molecular graph or hydrogen-suppressed molecular graph is the molecular graph with hydrogen vertices deleted.

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.

Prehistory
Arthur Cayley was probably the first to publish results that consider molecular graphs as early as in 1874, even before the introdution of the term "graph". For the purposes of enumeration of isomers, 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 Scottich 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 fiollowing century.