# Definition:Path Graph

## Definition

A **path graph** is a tree which has a path which passes through all its vertices.

The path graph with $n$ vertices is denoted $P_n$.

## Examples

## Basic Properties

- $P_1$ is the edgeless graph $N_1$, and also the complete graph $K_1$.

- $P_1$ is (trivially) Hamiltonian and Eulerian.

- $P_2$ is the complete bipartite graph $K_{1, 1}$, and is $1$-regular.

- $P_3$ is the complete bipartite graph $K_{1, 2}$.

- $P_n$ is semi-Hamiltonian and semi-Eulerian for all $n \ge 2$.

- $P_n$ is bipartite for all $n$.

- $P_n$ is a tree for all $n$.