  In this paper, we deal with a generalization $\Gamma(\Omega,q)$ of the
bipartite graphs $D(k,q)$ proposed by Lazebnik and Ustimenko, where $\Omega$ is
a set of binary sequences that are adopted to index the entries of the
vertices. A few sufficient conditions on $\Omega$ for $\Gamma(\Omega,q)$ to
admit a variety of automorphisms are proposed. A sufficient condition for
$\Gamma(\Omega,q)$ to be edge-transitive is proposed further. A lower bound of
the number of the connected components of $\Gamma(\Omega,q)$ is given by
showing some invariants for the components. For $\Gamma(\Omega,q)$, paths and
cycles which contain vertices of some specified form are investigated in
details. Some lower bounds for the girth of $\Gamma(\Omega,q)$ are then shown.
In particular, one can give very simple conditions on the index set $\Omega$ so
as to assure the generalized graphs $\Gamma(\Omega,q)$ to be a family of graphs
with large girth.
