# Why does Graph Theory Use the term Edge(s) to Describe Connectors (and not Line(s))?

Any insights into the historical rationale for using the term 'Edges' to describe connections between vertices...as distinct from the term 'Lines'?

• In directed graphs, they're arrows not lines. Sep 26 at 2:16
• The professor who gave me an unudergrad course on graph theory told us that it would be possible to write at least three introductory books on graph theory such that they would not share a single technical term. Didn't check, but it's plausible :-). I use the story whenever I encounter people (FBers are worse than the askers here) who think their terminology (on any math topic) is the canonical one. Sep 26 at 4:17

Historically - and not all that long ago! - terminology in graph theory was much more varied, and you could expect to see a graph defined with "points" and "lines" just as often as with "vertices" and "edges". Other options, like "nodes" and "arcs", survive to this day.

An interesting example is Frank Harary's Graph Theory, a textbook published in 1969. Harary uses "points" and "lines" when discussing abstract graphs, and "vertices" and "edges" for geometrically defined graphs, such as for plane embeddings of a graph. Harary's justification for this is that "points" and "lines" are already used in the study of incidence structures, of which graphs are a special case. It's also easy to see where "vertices" and "edges" come from: the study of polyhedral graphs.

I think that graph theory has moved away from "points" and "lines" precisely because they're used in many different ways in different contexts; on the other hand, nobody else uses "vertex" or "edge" to mean anything other than what the graph theorists mean. Back to the study of incidence structures: suppose you're studying the Fano plane, with its $$7$$ points and $$7$$ lines, each line containing $$3$$ points. You would very much like to define its incidence bigraph: the bipartite graph with

• $$14$$ vertices, one for every point and line of the Fano plane;
• $$21$$ edges: every time a point $$P$$ of the Fano plane lies on a line $$\ell$$, there is an edge between the two corresponding vertices of the incidence bigraph.

Imagine trying to make this definition when your bipartite graph is also made up of "points" and "lines"!

A descendant of "point" and "line" terminology still survives in modern graph theory, in the term line graph. (In the "point" and "line" terminology, it's easy to see why the line graph is called the line graph: its points are the lines of $$G$$!)

The adoption of the term "edges" to describe connections between vertices in graph theory can be traced back to the pioneering work of Leonhard Euler in the 18th century. Euler's publication of "Solutio problematis ad geometriam situs pertinentis" in 1736 marked a significant milestone in the development of graph theory.

In his paper, Euler introduced the concept of a graph as a mathematical abstraction to represent relationships between objects. He used the term "lines" to describe the connections between points or vertices in his graphs. However, it was later mathematicians and researchers who popularized the term "edges" as a more suitable and intuitive descriptor for these connections.

One notable figure in this process was August Ferdinand Möbius, a German mathematician who made significant contributions to graph theory in the 19th century. Möbius advocated for the term "Kanten" (meaning "edges" in German) in his writings, which had a lasting impact on the terminology used in graph theory.

The term "edges" gained further prominence through the works of other influential mathematicians and computer scientists. For example, in his influential book "Graph Theory" published in 1959, Frank Harary extensively used the term "edges" to describe the connections between vertices in graphs.

Over time, the term "edges" became widely accepted and established as the standard terminology in graph theory. Its usage has been pervasive across various branches of mathematics, computer science, and related fields.

As to the actual reason for why it was called "edges" (correct me if I am wrong) but I suppose it's because of the mathematical definition of a graph.

Hope it helps! Have a great day!

Edit 2: As Kurt G. pointed out, it would be quite interesting to mention the Königsberg Bridge Problem, also known as the Seven Bridges of Königsberg, a famous mathematical puzzle that was solved by Euler. The problem involves the city of Königsberg (now Kaliningrad, Russia), which was divided by the Pregel River into four land masses connected by seven bridges. The challenge was to find a walk through the city that would cross each of the seven bridges exactly once and return to the starting point. Euler approached this problem by abstracting it and representing the land masses as points and the bridges as lines connecting them, thus creating what is known as a graph.

• Very important to mention this in connection with L. Euler. Sep 25 at 16:42
• Interesting - I used Harary's textbook as my source for the use of "points" and "lines", after finding no older English-language graph theory textbook to reference. But it's true that at least in the planar graphs chapter, Harary does end up saying "vertices" and "edges" a lot, and it would be ironic if that was a major reason why we ended up using them for all graphs. Sep 25 at 16:46
• While 'edges' is a direct translation of the German 'Kanten', the common German word for vertices is 'Ecken' which translates to corners. This fits with the polyhedral graph origin but can be very confusing when translating because one goes from a German graph $(E,K)$ to an English graph $(V,E)$. Sep 26 at 5:36
• @quarague I am a proponent for using the word Verbindungen (‘connections’) when talking about graphs in German. But to reduce confusion, use the English letters for the sets (i.e. Ecken $V$ und Verbindungen $E$). Sep 26 at 7:44
• Also, "The adoption of the term "edges" to describe connections between vertices in graph theory can be traced back to the pioneering work of Leonhard Euler[...] In his paper, Euler introduced the concept of a graph as a mathematical abstraction to represent relationships between objects. He used the term "lines" to describe the connections between points or vertices in his graphs. However, it was later mathematicians and researchers who popularized the term 'edges'" Sep 27 at 23:34