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In the mathematical field of graph theory, a **path graph** (or **linear graph**) is a graph whose vertices can be listed in the order *v*_{1}, *v*_{2}, ..., *v _{n}* such that the edges are {

Path graph | |
---|---|

Vertices | n |

Edges | n − 1 |

Radius | ⌊n/2⌋ |

Diameter | n − 1 |

Automorphisms | 2 |

Chromatic number | 2 |

Chromatic index | 2 |

Spectrum | |

Properties | Unit distance Bipartite graph Tree |

Notation | P_{n}^{[1]} |

Table of graphs and parameters |

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).

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.

**^**While it is most common to use P_{n}for a path of n vertices, some authors (e.g. Diestel) use P_{n}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.

- Weisstein, Eric W. "Path Graph".
*MathWorld*.