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 (cf. Yadav et al. 2024), independence polynomial,
and reliability polynomial of the ladder
graph 
are given by
 |  |  |
(1)
|
 |  | ![2^(-(n+1))[(s-3x-1)(x-s+1)^n+(s+3x+1)(x+s+1)^n]](/images/equations/LadderGraph/Inline17.svg) |
(2)
|
 |  | ![((p-1)^(2n-1))/(2^nsqrt(p(9p+2)+1))[(3p-sqrt(p(9p+2)+1)+1)^n-(3p+sqrt(p(9p+2)+1)+1)^n],](/images/equations/LadderGraph/Inline20.svg) |
(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)
|
." J. Appl. Math. Comput., 2024.