Embedding the vertices of arbitrary graphs into trees while minimizing some measure of overlap is an important problem with applications in computer science and physics. In this work, we consider the problem of bijectively embedding the vertices of an $n$-vertex graph $G$ into the \textit{leaves} of an $n$-leaf \textit{rooted binary tree} $\mathcal{T}$. The congestion of such an embedding is given by the largest size of the cut induced by the two components obtained by deleting any vertex of $\mathcal{T}$. We show that for any embedding, the congestion lies between $λ_2(G)\cdot 2n/9$ and $λ_n(G)\cdot n/4$, letting $0=λ_1(G)\le \cdots \le λ_n(G)$ be the Laplacian eigenvalues of $G$, and there is an embedding for which the congestion is at most $λ_n(G)\cdot 2n/9$. Beyond these general bounds, we determine the congestion exactly for hypercubes and lattice graphs, and obtain asymptotically tight bounds for random regular graphs and Erdős-Rényi graphs. We further introduce an efficient contraction procedure based on spectral ordering and dynamic programming, which produces low-congestion embeddings in practice. Numerical experiments on structured graphs, random graphs, and tensor network representations of quantum circuits validate our theoretical bounds and demonstrate the effectiveness of the proposed method. These results yield new spectral bounds on the memory and time complexity of exact tensor network contraction in terms of the underlying graph structure.

翻译：将任意图的顶点嵌入树中并最小化重叠度量是计算机科学和物理学中的重要问题。本文考虑将$n$个顶点图$G$的顶点双射嵌入到$n$个叶子的\textit{有根二叉树}$\mathcal{T}$的\textit{叶子}中的问题。此类嵌入的拥塞定义为删除$\mathcal{T}$的任意顶点后所得两个分支诱导的割的最大规模。我们证明，对于任何嵌入，拥塞介于$λ_2(G)\cdot 2n/9$与$λ_n(G)\cdot n/4$之间，其中$0=λ_1(G)\le \cdots \le λ_n(G)$为$G$的拉普拉斯特征值，且存在一个嵌入使拥塞至多为$λ_n(G)\cdot 2n/9$。除这些一般界外，我们精确确定了超立方体和网格图的拥塞，并得到了随机正则图和Erdős-Rényi图的渐近紧界。进一步，我们引入了一种基于谱序和动态规划的高效收缩方法，可在实践中生成低拥塞嵌入。关于结构化图、随机图以及量子电路张量网络表示上的数值实验验证了我们的理论界，并展示了所提方法的有效性。这些结果基于底图结构给出了精确张量网络收缩在内存和时间复杂度上的新谱界。