  The trees, respectively unicyclic graphs, on $n$ vertices with the smallest
Laplacian permanent are studied. In this paper, by edge-grafting
transformations, the $n$-vertex trees of given bipartition having the second
and third smallest Laplacian permanent are identified. Similarly, the
$n$-vertex bipartite unicyclic graphs of given bipartition having the first,
second and third smallest Laplacian permanent are characterized. Consequently,
the $n$-vertex bipartite unicyclic graphs with the first, second and third
smallest Laplacian permanent are determined.
