We investigate the relationship between the lexicographic product of graphs and their multi-word-representation number. Although the lexicographic product of two word-representable graphs need not itself be word-representable, a precise characterization has not previously been established. We provide a complete characterization, showing that for word-representable graphs $G_1$ and $G_2$, the lexicographic product $G_1 \circ G_2$ is word-representable if and only if $G_2$ is a comparability graph. For lexicographic powers, we prove that $G^{[k]}$ is word-representable if and only if $G$ is a comparability graph. The multi-word-representation number $μ$ for lexicographic powers and products satisfies the following bounds. If $G$ is a non-comparability graph, then $μ(G^{[k]}) \le k$, whereas if $G$ is the union of two comparability graphs, then $μ(G^{[k]}) = 2$. More generally, for graphs $G_1$ and $G_2$ with $μ(G_1) = k_1$ and $μ(G_2) = k_2$, the lexicographic product $H = G_1 \circ G_2$ satisfies the upper bound $μ(H) \le k_1 + k_2$. This bound is tight, with equality $μ(H) = k_1$, when $k_1 \ge k_2$ and $G_2$ is the union of $k_1$ comparability graphs. Moreover, if $G_1$ and $G_2$ are minimal non-word-representable graphs, then $μ(G_1 \circ G_2) \le 3$. Finally, we study the function $τ(n)$, which measures the size of the largest word-representable induced subgraph guaranteed in every $n$-vertex graph. By constructing extremal graphs via lexicographic powers, we establish a sublinear upper bound, showing that $τ(n) \le n^{0.86}$ for sufficiently large $n$.

翻译：我们研究了图的字典序积与其多词表示数之间的关系。尽管两个字可表示图的字典序积本身未必是词可表示的，但此前尚未建立精确的特征刻画。我们给出了完整刻画，表明对于词可表示图$G_1$和$G_2$，其字典序积$G_1 \circ G_2$是词可表示的当且仅当$G_2$是可比较图。对于字典序幂，我们证明$G^{[k]}$是词可表示的当且仅当$G$是可比较图。字典序幂与积的多词表示数$μ$满足以下界限：若$G$是非可比较图，则$μ(G^{[k]}) \le k$；若$G$是两个可比较图的并，则$μ(G^{[k]}) = 2$。更一般地，对于满足$μ(G_1) = k_1$和$μ(G_2) = k_2$的图$G_1$和$G_2$，其字典序积$H = G_1 \circ G_2$满足上界$μ(H) \le k_1 + k_2$。当$k_1 \ge k_2$且$G_2$是$k_1$个可比较图的并时，该界是紧的，即等式$μ(H) = k_1$成立。此外，若$G_1$和$G_2$均为极小非词可表示图，则$μ(G_1 \circ G_2) \le 3$。最后，我们研究了函数$τ(n)$，它衡量任意$n$顶点图中必然存在的最大词可表示诱导子图的大小。通过利用字典序幂构造极值图，我们建立了次线性上界，表明对于足够大的$n$，有$τ(n) \le n^{0.86}$。