The multi-word-representation number $μ(G)$ of a graph $G$ is the minimum number of word-representable graphs whose union is $G$. We study the behavior of $μ$ under six standard graph products: the lexicographic, Cartesian, rooted, corona, tensor, and strong products. For the Cartesian and rooted products, we show that $μ(G_1 \square G_2)=μ(G_1 \diamond G_2)=\max\{μ(G_1),μ(G_2)\}$. For the corona product, we prove that $μ(G_1 \odot G_2)\le \max\{μ(G_1),μ(G_2)\}+1$, and we identify a condition under which equality holds. For the lexicographic product, we establish $μ(G_1 \circ G_2)\le μ(G_1)+μ(G_2)$, which reduces to $\max\{μ(G_1),μ(G_2)\}$ under a comparability cover condition on $G_2$, and we characterize the case when the lexicographic product of two minimal non-word-representable graphs has $μ=2$. For the tensor product $G_1 \times G_2$, we show $μ(G_1 \times G_2)\le \log_3(\min\{χ(G_1),χ(G_2)\})$. For the strong product $G_1 \boxtimes G_2$, we establish $\max\{μ(G_1),μ(G_2)\}\le μ(G_1 \boxtimes G_2)\le \max\{μ(G_1),μ(G_2)\}+\log_3(\min\{χ(G_1),χ(G_2)\})$. For lexicographic powers $G^{[k]}$, we prove that $μ(G^{[k]})\le k$ when $G$ is word-representable but not a comparability graph, and in general $μ(G^{[k]})$ is bounded by the comparability cover number of $G$. We further show that $G^{[k]}$ is word-representable if and only if $G$ is a comparability graph. As an application, we obtain a sublinear upper bound on the extremal function $τ(n)$, defined as the largest integer such that every $n$-vertex graph contains a word-representable induced subgraph of that size; in particular, $τ(8^k)\le 6^k$, implying $τ(n)\le n^{\log_8 6+ε}$ for large $n$.

翻译：图G的`多词可表示数`μ(G)是覆盖G所需的词可表示图的最小个数。本文研究μ在六种标准图乘积（词典序乘积、笛卡尔乘积、有根乘积、冠状乘积、张量乘积与强乘积）下的行为。对于笛卡尔乘积与有根乘积，我们证明μ(G₁□G₂)=μ(G₁◇G₂)=max{μ(G₁),μ(G₂)}。对于冠状乘积，我们证明μ(G₁⊙G₂)≤max{μ(G₁),μ(G₂)}+1，并给出等式成立的条件。对于词典序乘积，我们建立μ(G₁∘G₂)≤μ(G₁)+μ(G₂)，当G₂满足可比覆盖条件时该上界可简化为max{μ(G₁),μ(G₂)}，并刻画了两个极小非词可表示图的词典序乘积μ=2的情形。关于张量乘积G₁×G₂，我们证明μ(G₁×G₂)≤log₃(min{χ(G₁),χ(G₂)})。关于强乘积G₁⊠G₂，我们建立max{μ(G₁),μ(G₂)}≤μ(G₁⊠G₂)≤max{μ(G₁),μ(G₂)}+log₃(min{χ(G₁),χ(G₂)})。对于词典序幂G^{[k]}，我们证明当G是词可表示但非可比图时μ(G^{[k]})≤k，且一般情形下μ(G^{[k]})受G的可比覆盖数界定。进一步证明G^{[k]}是词可表示图当且仅当G是可比图。作为应用，我们得到极值函数τ(n)（定义为任意n顶点图中最大词可表示诱导子图规模）的次线性上界：特别地，τ(8^k)≤6^k，从而对充分大的n有τ(n)≤n^{log₈6+ε}。