Given a graph $G = (V, E)$ a path is a sequence of edges $\{e_i = (x_i, y_i)\}_{i=1}^k$ such that $x_i = y_{i+1}$ for all $1 \leq i < k$. We say the length of the path is $k$.

Visual representation

Lets use our simple graph below simple_path There is a path that goes $(1,3), (3,4)$ from vertex 1 to vertex 4. Note there is an implicit directionality when we say this.

The path graph

In graph theory we define the Path graph $P_k$ to be a graph as a graph with

$$V = \{1,2, \ldots, k\} \ \mbox{ and } \ E = \{(i,i+1)\ \vert \ 1 \leq i < k \}.$$

There is also an infinite analogy, both one sided and two sided.