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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)

Cocktail Party Graph, Crossed Prism Graph, Cycle Graph, Cyclotomic Graph, Gear Graph, Grid Graph, Hadamard Graph, Helm Graph, Ladder Rung Graph, Möbius Ladder, Path Graph, Prism Graph, Web Graph, Wheel Graph

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## 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.