Ladder Graph

The -ladder graph can be defined as , where  is a path graph (Hosoya and Harary 1993; Noy and Ribó 2004, Fig. 1). It is therefore equivalent to the  grid graph. The ladder graph is named for its resemblance to a ladder consisting of two rails and  rungs between them (though starting immediately at the bottom and finishing at the top with no offset).

Hosoya and Harary (1993) also use the term "ladder graph" for the graph Cartesian product , where  is the complete graph on two nodes and  is the cycle graph on  nodes. This class of graph is however more commonly known as a prism graph.

Ball and Coxeter (1987, pp. 277-278) use the term "ladder graph" to refer to the graph known in this work as the ladder rung graph.

The ladder graph  is graceful (Maheo 1980).

The chromatic polynomial, independence polynomial, and reliability polynomial of the ladder graph  are given by

			(1)
			(2)
			(3)

where . Recurrence equations for the chromatic polynomial, independence polynomial, matching polynomial, rank polynomial, and reliability polynomial are given by

			(4)
			(5)
			(6)
			(7)
			(8)

REFERENCES

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, 1987.

Hosoya, H. and Harary, F. "On the Matching Properties of Three Fence Graphs." J. Math. Chem. 12, 211-218, 1993.

Maheo, M. "Strongly Graceful Graphs." Disc. Math. 29, 39-46, 1980.

Noy, M. and Ribó, A. "Recursively Constructible Families of Graphs." Adv. Appl. Math. 32, 350-363, 2004.