  A pair $(A,B)$ of square $(0,1)$-matrices is called a \emph{Lehman pair} if
$AB^T=J+kI$ for some integer $k\in\{-1,1,2,3,\ldots\}$. In this case $A$ and
$B$ are called \emph{Lehman matrices}. This terminology arises because Lehman
showed that the rows with the fewest ones in any non-degenerate minimally
nonideal (mni) matrix $M$ form a square Lehman submatrix of $M$. Lehman
matrices with $k=-1$ are essentially equivalent to \emph{partitionable graphs}
(also known as $(\alpha,\omega)$-graphs), so have been heavily studied as part
of attempts to directly classify minimal imperfect graphs. In this paper, we
view a Lehman matrix as the bipartite adjacency matrix of a regular bipartite
graph, focusing in particular on the case where the graph is cubic. From this
perspective, we identify two constructions that generate cubic Lehman graphs
from smaller Lehman graphs. The most prolific of these constructions involves
repeatedly replacing suitable pairs of edges with a particular $6$-vertex
subgraph that we call a $3$-rung ladder segment. Two decades ago, L\"{u}tolf \&
Margot initiated a computational study of mni matrices and constructed a
catalogue containing (among other things) a listing of all cubic Lehman
matrices with $k =1$ of order up to $17 \times 17$. We verify their catalogue
(which has just one omission), and extend the computational results to $20
\times 20$ matrices. Of the $908$ cubic Lehman matrices (with $k=1$) of order
up to $20 \times 20$, only two do not arise from our $3$-rung ladder
construction. However these exceptions can be derived from our second
construction, and so our two constructions cover all known cubic Lehman
matrices with $k=1$.
