A graph $G(V, E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that for distinct letters $x,y\in V$, $x$ and $y$ alternate in $w$ if and only if they are adjacent in $G$. In general, determining whether a graph is word-representable is an NP-complete problem. A graph is co-bipartite if its complement is bipartite. Therefore, the vertex set of a co-bipartite graph can be partitioned into two disjoint subsets $X$ and $Y$ such that the subgraphs induced by $X$ and $Y$ are cliques. In this paper, we obtain necessary and sufficient conditions for a co-bipartite graph to be word-representable in terms of a vertex ordering. Based on this ordering, we study the representation number of word-representable co-bipartite graphs and analyse the speed and entropy of this graph class. We show that the representation number of any word-representable co-bipartite graph is at most $3$, and that permutation graphs are the only co-bipartite graphs with representation number $2$. We prove that the speed is $2^{O(n \log n)}$ and the entropy is $0$. This provides an asymptotic bound on the number of labelled graphs in this class, which is significantly smaller than the known bound for the class of all co-bipartite graphs. These results provide a better understanding of the structure and enumeration of word-representable co-bipartite graphs and show that vertex ordering is an effective tool for studying this class.

翻译：图 $G(V, E)$ 称为可词表示的，若存在一个字母表 $V$ 上的单词 $w$，使得对于不同的字母 $x,y\in V$，$x$ 与 $y$ 在 $w$ 中交替出现当且仅当它们在 $G$ 中相邻。一般而言，判定一个图是否可词表示是 NP 完全问题。若一个图的补图是二部图，则该图称为共二部图。因此，共二部图的顶点集可划分为两个互不相交的子集 $X$ 和 $Y$，使得 $X$ 和 $Y$ 诱导的子图均为团。本文中，我们得到了共二部图可词表示的充要条件，该条件以顶点排序的形式给出。基于此排序，我们研究了可词表示共二部图的表示数，并分析了该类图的速度与熵。我们证明：任何可词表示共二部图的表示数至多为 $3$，且置换图是表示数为 $2$ 的唯一共二部图。我们证明其速度为 $2^{O(n \log n)}$，熵为 $0$。这给出了该类图中可标记图的渐近上界，该上界显著小于已知的全体共二部图类的界。这些结果加深了对可词表示共二部图结构与计数的理解，并表明顶点排序是研究该类图的有效工具。