Path graph

---

title: "Path graph" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["trees-(graph-theory)", "parametric-families-of-graphs"] description: "Graph with nodes connected linearly" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Path_graph" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Graph with nodes connected linearly ::

::data[format=table title="infobox graph"]

Field	Value
name	Path graph
image	[[Image:Path-graph.svg
image_caption	A path graph on 6 vertices
vertices	n
edges	n − 1
automorphisms	2
diameter	n − 1
radius	⌊n/2⌋
chromatic_number	2
chromatic_index	2
spectrum	{2\cos\left(\frac{k\pi}{n+1}\right);k=1,\ldots,n}
properties	Unit distance
Bipartite graph
Tree
notation	Pn
::

| name = Path graph | image = [[Image:Path-graph.svg|250px]] | image_caption = A path graph on 6 vertices | vertices = n | edges = n − 1 | automorphisms = 2 | diameter = n − 1 | radius = ⌊n/2⌋ | chromatic_number = 2 | chromatic_index = 2 | spectrum = {2\cos\left(\frac{k\pi}{n+1}\right);k=1,\ldots,n} | properties = Unit distance Bipartite graph Tree | notation = Pn

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, v**i+1} where . 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.

References

• {{cite book |author1 = Bondy, J. A. |author-link = John Adrian Bondy |author2 = Murty, U. S. R. |author2-link = U. S. R. Murty |title = Graph Theory with Applications |year = 1976 |publisher = North Holland |isbn = 0-444-19451-7 |pages = 12–21 |url = https://archive.org/details/graphtheorywitha0000bond/page/12 |url-status =
• {{cite book | author = Diestel, Reinhard | author-link = Reinhard Diestel | title = Graph Theory | edition = 3rd | url = http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/ | publisher = Graduate Texts in Mathematics, vol. 173, Springer-Verlag | year = 2005 | isbn = 3-540-26182-6 | pages = 6–9}}

References

1. While it is most common to use {{mvar. P_n for a path of {{mvar. n vertices, some authors (e.g. Diestel) use {{mvar. P_n for a path of {{mvar. n ''edges'' and {{math. ''n''+1 vertices.

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