Structural properties of biclique graphs and the distance formula

A \textit{biclique} is a maximal induced complete bipartite subgraph of $G$. The \textit{biclique graph} of a graph $G$, denoted by $KB(G)$, is the intersection graph of the family of all bicliques of $G$. In this work we study some structural properties of biclique graphs which are necessary conditions for a graph to be a biclique graph. In particular, we prove that for biclique graphs that are neither a $K_3$ nor a \textit{diamond}, the number of vertices of degree $2$ is less than half the number of vertices in the graph. Also, we present forbidden structures. For this, we introduce a natural definition of the distance between bicliques in a graph. We give a formula that relates the distance between bicliques in a graph $G$ and the distance between their respective vertices in $KB(G)$. Using these results, we can prove not only this new necessary condition involving the degree, but also that some graphs are not biclique graphs. For example, we show that the \textit{crown} is the smallest graph that is not a biclique graph although the known necessary condition for biclique graphs holds, answering an open problem about biclique graphs. Finally, we present some interesting related conjectures and open problems.