Log-concavity of the independence polynomials of Wₚ graphs

Let G be a Wₚ graph if n≥ p and every p pairwise disjoint independent sets of G are contained within p pairwise disjoint maximum independent sets. In this paper, we establish that every Wₚ graph G is p-quasi-regularizable if and only if n≥ (p+1)α, where α is the independence number of G. This finding ensures that the independence polynomial of a connected Wₚ graph G is log-concave whenever (p+1)α ≤ n≤ 2pα +p+1. Furthermore, we demonstrate that the independence polynomial of the clique corona G∘ Kₚ is invariably log-concave for all p≥ 1. As an application, we validate a long-standing conjecture claiming that the independence polynomial of a very well-covered graph is unimodal.