On a Generalization of the Bipartite Graph $D(k,q)$

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.