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

TL;DR

This paper proves a conjecture relating the structure of $ ext{W}_2$ graphs to their $2$-quasi-regularizability and explores conditions for the log-concavity and unimodality of their independence polynomials.

Contribution

It establishes the equivalence between $2$-quasi-regularizability and a specific inequality involving graph parameters, and provides criteria for independence polynomial properties in $ ext{W}_p$ graphs.

Findings

Connected $ ext{W}_2$ graphs are $2$-quasi-regularizable iff $n(G) \\ge 3\\alpha(G)$.

Local expansion theorem: non-maximum independent sets satisfy $|N_G(A)| \\ge 2|A|$.

Explicit regions for log-concavity and unimodality of independence polynomials in $ ext{W}_2$ graphs.

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.