双团图的结构性质及距离公式
Structural properties of biclique graphs and the distance formula
摘要：
\textit{biclique}是$G$的极大导出完全二部子图。图$G$的\textit{biclique graph}，记作$KB(G)$，是$G$的所有双结点族的交图。本文研究了图的结构性质，这些性质是图是图的必要条件。特别是，我们证明了，对于biclique图既不是一个$K_3$也不是一个\textit{diamond}，度$2$的顶点数是小于一半的顶点数的图。此外，我们提出了禁止结构。为此，我们引入一个自然的定义之间的距离bicliques在一个图。本文给出了一个关系图$G$中两点间距离与$KB(G)$中两点间距离的公式。利用这些结果，我们不仅可以证明这个新的涉及度的必要条件，而且还可以证明某些图不是双团图。例如，我们表明，\textit{crown}是最小的图，不是一个biclique图，虽然已知的必要条件biclique图举行，回答一个开放的问题biclique图。最后，我们提出了一些有趣的相关架构和开放的问题。
英文摘要：
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.
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