Log-concavity of the independence polynomials of W_p graphs

Let G be a graph of order n . For a positive integer p , G is said to be a W_p 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 connected W_p graph G is p -quasi-regularizable if and only if n≥(p+1)⋅α, where α is the independence number of G and p≠2. This finding ensures that the independence polynomial of a connected W_p graph G is log-concave whenever (p+1)⋅α≤n≤p⋅α+2p⋅α+p and α²4(α+1)≤p, or p⋅α+2p⋅α+p<n≤(α²+1)⋅p+(α−1)²α−1 and α(α−1)α+1≤p. Moreover, the clique corona graph G∘K_p serves as an example of the W_p graph class. We further demonstrate that the independence polynomial of G∘K_p is always log-concave for sufficiently large p .