Log-concavity of the independence polynomials of $\mathbf{W}_{p}$ graphs

Abstract

Let $G$ be a graph of order $n$. For a positive integer $p$, $G$ is said to be a $\mathbf{W}_{p}$ graph if $n\geq p$ 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 $\mathbf{W}_{p}$ graph $G$ is $p$-quasi-regularizable if and only if $n\geq(p+1)\cdot\alpha$, where $\alpha$ is the independence number of $G$ and $p\neq2$. This finding ensures that the independence polynomial of a connected $\mathbf{W}_{p}$ graph $G$ is log-concave whenever $(p+1)\cdot\alpha\leq n\leq p\cdot\alpha+2\sqrt{p\cdot\alpha+p}$ and $\frac{\alpha^{2}}{4\left( \alpha+1\right) }\leq p$, or $p\cdot\alpha+2\sqrt{p\cdot\alpha+p}<n\leq \frac{\left( \alpha^{2}+1\right) \cdot p+\left( \alpha-1\right) ^{2}}{\alpha-1}$ and $\frac{\alpha\left( \alpha-1\right) }{\alpha+1}\leq p$. Moreover, the clique corona graph $G\circ K_{p}$ serves as an example of the $\mathbf{W}_{p}$ graph class. We further demonstrate that the independence polynomial of $G\circ K_{p}$ is always log-concave for sufficiently large $p$. Keywords: very well-covered graph; quasi-regularizable graph; corona graph; $\mathbf{W}_{p}$ graph; independence polynomial; log-concavity.