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

Hoang, Levit, Mandrescu and Pham asked for structural conditions ensuring that the independence polynomial of a ${\text{W}}_p$ graph is log-concave, or at least unimodal, and conjectured that a connected ${\text{W}}_2$ 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 ${\text{W}}_2$, then every non-maximum independent set $A$ satisfies $$\mathrm{\mid }{N}_G(A)\mathrm{\mid }\ge 2\mathrm{\mid }A\mathrm{\mid }\mathrm{.}$$ Thus the only possible obstruction to $2$-quasi-regularizability in a connected ${\text{W}}_2$ 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 ${\text{W}}_p$ 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 ${\text{W}}_2$ graphs.