A path graph has vertices that can be listed in order $v_1, v_2, …,v_n$ such that edges are ${v_i, v_{i+i}}$ where $i = 1, 2, …, n-1$. That is rather complicated definition for a simple concept that is easily visualized. The image below represents a path graph.

There is only one path through the graph and each vertex is connected to the preceding vertex. By this definition, there will always be $n-1$ edges.