+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Graph theory +-- {: .hide} [[!include graph theory - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea In just about any concept of [[graph]], an **edge** is something which *connects two [[vertex|vertices]]*. In a [[directed graph]] an edge is directed, whereas in an undirected graph (the default meaning of "graph" in contemporary combinatorics) it is not. ## Terminological remarks Edges are sometimes called **arcs** (less common in combinatorics, but sometimes used in other fields) or **lines** (e.g. [Harary, Chapter 2](#HararyGraphTheory)). The latter is now unusual and considered old-fashioned, but survives in the standard technical term [[line graph]] (which is more common than "edge graph" for the same concept). In more specialized contexts other terms are used. For instance, if the [[graph]] is the [[quiver]] underlying a [[category]], then an edge is a [[morphism]]; while if it underlies a [[simplicial set]], then an edge is a 1-[[dimension|dimensional]] [[simplex]]. Historically, the use of "edge" for connections even in abstract graphs probably derives from the natural connections between [[graph theory]] and the theory of [[polyhedra]]. One such connection is [[Ernst Steinitz|Steinitz's]] [[Steinitz's theorem|1916 theorem]] on the one-dimensional skeleta of [[polyhedra]] in three-dimensional [[Euclidean space]]. However, some deplore the geometric connotations of "edge", with its geometric connotations of straightness and rigidity. ## Related concepts * [[graph]], [[directed graph]], [[quiver]], [[digraph]] * [[dimer]] * [[vertex]], [[simplex]], [[dendrex]] ## References * Frank Harary: Graph Theory. Addison-Wesley. 1969 {#HararyGraphTheory} [[!redirects edge]] [[!redirects edges]] [[!redirects arc]] [[!redirects arcs]]