Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Chen, Jungang, Xie, Jiaxin
Institution: Xiamen University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214773041