path graph

{{short description|Graph with nodes connected linearly}}

{{about|a family of graphs|paths as parts of arbitrary graphs|Path (graph theory)}}

{{distinguish|line graph}}

{{infobox graph

| name = Path graph

| image = 250px

| image_caption = A path graph on 6 vertices

| vertices = {{mvar|n}}

| edges = {{math|n − 1}}

| automorphisms = 2

| diameter = {{math|n − 1}}

| radius = {{math|⌊n/2⌋}}

| chromatic_number = 2

| chromatic_index = 2

| spectrum = $\backslash \{2\backslash cos\backslash left(\backslash frac\{k\backslash pi\}\{n+1\}\backslash right);$$k=1,\backslash ldots,n\backslash \}$

| properties = Unit distance
Bipartite graph
Tree

| notation = {{mvar|P_n}}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.

}}

In the mathematical field of graph theory, a path graph (or linear graph) is a graph whose vertices can be listed in the order {{math|v₁, v₂, ..., v_n}} such that the edges are {{math|{v_i, v_i+1}}} where {{math|1=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).