Quantitative Merino--Welsh inequalities for joins

Source abstract

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.