Definition:Path Graph

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

 * PathGraphs.png

Basic Properties

 * $$P_1$$ is the null 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 $$n \ge 2$$.