Path graph

This article is about a family of graphs. For paths as parts of arbitrary graphs, see Path (graph theory).
Not to be confused with line graph.
Path graph
A path graph on 6 vertices
Verticesn
Edgesn − 1
Radius⌊n/2⌋
Diametern − 1
Automorphisms2
Chromatic number2
Chromatic index2
Spectrum
PropertiesUnit distance
Bipartite graph
Tree
NotationPn[1]
Table of graphs and parameters

In the mathematical field of graph theory, a path graph (or linear graph) is a graph whose vertices can be listed in the order v1, v2, ..., vn such that the edges are {vi, vi+1} where i = 1, 2, ..., n − 1. Equivalently, a path with at least two vertices is connected and has two terminal vertices (vertices of degree 1), while all others (if any) have degree 2.

Paths are often important in their role as subgraphs of other graphs, in which case they are called paths in that graph. A path is a particularly simple example of a tree, and in fact the paths are exactly the trees in which no vertex has degree 3 or more. A disjoint union of paths is called a linear forest.

Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts. See, for example, Bondy and Murty (1976), Gibbons (1985), or Diestel (2005).

As Dynkin diagrams

In algebra, path graphs appear as the Dynkin diagrams of type A. As such, they classify the root system of type A and the Weyl group of type A, which is the symmetric group.

See also

References

  1. ^ While it is most common to use Pn for a path of n vertices, some authors (e.g. Diestel) use Pn for a path of n edges and n+1 vertices.
  • Bondy, J. A.; Murty, U. S. R. (1976). Graph Theory with Applications. North Holland. pp. 12–21. ISBN 0-444-19451-7.
  • Diestel, Reinhard (2005). Graph Theory (3rd ed.). Graduate Texts in Mathematics, vol. 173, Springer-Verlag. pp. 6–9. ISBN 3-540-26182-6.

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