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arXiv · 2605.14076

The 2-Quasi-Regularizability Conjecture and Independence Polynomials of Wp Graphs

Kevin Pereyra
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Abstract

Hoang, Levit, Mandrescu and Pham asked for structural conditions ensuring that the independence polynomial of a $\W_p$ graph is log-concave, or at least unimodal, and conjectured that a connected $\W_2$ graph is $2$-quasi-regularizable if and only if $n(G)\ge 3\alpha(G)$ (2026). We prove the conjecture. The key point is a local expansion theorem: if $G$ is connected and belongs to $\W_2$, then every non-maximum independent set $A$ satisfies \[ |N_G(A)|\ge 2|A|. \] Thus the only possible obstruction to $2$-quasi-regularizability in a connected $\W_2$ graph comes from maximum independent sets, where the condition is exactly $n(G)-\alpha(G)\ge 2\alpha(G)$. We also give coefficient criteria for log-concavity and unimodality of independence polynomials of $\W_p$ graphs. These criteria combine the standard two-sided coefficient inequalities, as collected by Hoang--Levit--Mandrescu--Pham, with $\lambda$-quasi-regularizability. The new $p=2$ threshold proved here inserts the missing case into the same framework and yields explicit log-concavity and unimodality regions for connected $\W_2$ graphs.

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Kevin Pereyra. 2026-05-13. The 2-Quasi-Regularizability Conjecture and Independence Polynomials of Wp Graphs. https://arxiv.org/abs/2605.14076

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