The path graph 
is a tree on 
vertices, with two nodes of vertex degree
1, and the other 
nodes of vertex degree
2. A path graph is therefore a graph that can be drawn
so that all of its vertices and edges
lie on a single straight line (Gross and Yellen 2006, p. 18).
The path graph on 
vertices is implemented in the Wolfram
Language as PathGraph[Range[n]],
and precomputed properties of path graphs are available as GraphData[
"Path", n
]. (Note that the Wolfram
Language believes cycle graphs to be path graphs,
a convention that seems neither standard nor useful.)
The path graph 
is known as the singleton graph and is equivalent
to the complete graph 
and the star graph 
. 
is isomorphic to the complete bipartite graph 
and 
to 
.
Path graphs 
are graceful.
The path graph 
has chromatic polynomial, independence
polynomial, matching polynomial, and reliability polynomial given by
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
where 
and 
is a Fibonacci polynomial.
These have recurrence equations
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
The line graph of 
is isomorphic to 
.
is the Cayley
graph of the permutations 
2,
1
and 
1, 3, 2
.
An explicit graph 
can be tested for being a path graph in the Wolfram
Language using PathGraphQ[g]
&& AcyclicGraphQ[g].
