Quantitative Merino--Welsh inequalities for joins
Abstract
For a connected graph G, let Q(G)=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≤ a≤ b, then Q(A∨ 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≥4, we prove Q(K2∨ F)≥ 27/n2(3/2)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.
Related
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BibTeX
@misc{chen2026quantitative,
title = {{Quantitative Merino--Welsh inequalities for joins}},
author = {Jungang Chen and Jiaxin Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33680},
eprint = {2609.33680},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33680},
}
RIS
TY - UNPB
TI - Quantitative Merino--Welsh inequalities for joins
AU - Jungang Chen
AU - Jiaxin Xie
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33680
UR - https://arxiv.org/abs/2609.33680
AB - For a connected graph G, let Q(G)=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≤ a≤ b, then Q(A∨ 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≥4, we prove Q(K_2∨ F)≥ 27/n^2(3/2)^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.
ER -