AI 中文总结
本文针对图并运算下的 Merino--Welsh 不等式，给出了定量下界，并证明至少三个非空因子的并图及无割边的完全多部图满足严格不等式，方法结合定向估计与生成树比较。
AI 中文摘要
对于连通图 \\(G\\)，设 \\(Q(G)=\frac{T(G;2,0)T(G;0,2)}{T(G;1,1)^2}\\)。我们在图并运算下获得 \\(Q\\) 的定量下界。若 \\(A\\) 和 \\(B\\) 是阶数满足 \\(3\le a\le b\\) 的任意简单图，则 \\(Q(A\vee B)\\) 有一个仅依赖于 \\(a\\) 和 \\(b\\) 的显式下界，且该下界严格大于 1。我们进一步量化了两个因子内部边所带来的改进。对于每个简单图 \\(F\\)，其中 \\(n=|V(F)|+2\ge4\\)，我们证明 \\(Q(K_2\vee F)\ge \frac{27}{n^2}\left(\frac32\right)^{n-4}\\)。因此，每个至少包含三个非空因子的并图，以及每个至少有一条边且无割边的完全多部图，都满足严格的乘法 Merino--Welsh 不等式。证明结合了定向估计与基于有效电阻和块消除的生成树比较。
英文摘要
For a connected graph \(G\), let \[ Q(G)=\frac{T(G;2,0)T(G;0,2)}{T(G;1,1)^2}. \] We obtain quantitative lower bounds for \(Q\) under the graph join operation. If \(A\) and \(B\) are arbitrary simple graphs of orders \(3\le a\le b\), then \(Q(A\vee B)\) admits an explicit lower bound depending only on \(a\) and \(b\), and this bound is strictly greater than \(1\). We further quantify the improvement produced by edges inside the two factors. For every simple graph \(F\), with \(n=|V(F)|+2\ge4\), we prove \[ Q(K_2\vee F)\ge \frac{27}{n^2}\left(\frac32\right)^{n-4}. \] Consequently, every join of at least three nonempty factors, and every complete multipartite graph with at least one edge and no cut edges, satisfies the strict multiplicative Merino--Welsh inequality. The proofs combine orientation estimates with spanning-tree comparisons based on effective resistance and block elimination.
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