Log-concavity of the independence polynomials of W₏ graphs

Let G be a W₏ 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 W₏ graph G is p-quasi-regularizable if and only if n (p+1), where is the independence number of G. This finding ensures that the independence polynomial of a connected W₏ graph G is log-concave whenever (p+1) n 2p +p+1. Furthermore, we demonstrate that the independence polynomial of the clique corona G K₏ 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.