Path Graph

The path graph  is a tree 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 of length  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 graph, 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 . These have recurrence equations

			(5)
			(6)
			(7)
			(8)

The line graph of  is isomorphic to .

 is the Cayley graph of the permutations 2, 1and 1, 3, 2.

REFERENCES

Gross, J. T. and Yellen, J. Graph Theory and Its Applications, 2nd ed. Boca Raton, FL: CRC Press, 2006.