Title: Log-concavity of the independence polynomials of $\mathbf{W}_{p}$ graphs Show Chinese title

Title: $\mathbf{W}_{p}$图的独立多项式的对数凹性

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. Show Chinese abstract

Abstract: 设$G$是一个阶为$n$的图。 对于正整数$p$，如果$n\geq p$且每个$p$两两不相交的独立集的$G$都包含在$p$两两不相交的最大独立集中，则称$G$是一个$\mathbf{W}_{p}$图。 在本文中，我们证明每个连通的$\mathbf{W}_{p}$图$G$是$p$-准正则可化当且仅当$n\geq(p+1)\cdot\alpha$，其中$\alpha$是$G$的独立数，且$p\neq2$。 这一发现确保了当$(p+1)\cdot\alpha\leq n\leq p\cdot\alpha+2\sqrt{p\cdot\alpha+p}$且$\frac{\alpha^{2}}{4\left( \alpha+1\right) }\leq p$，或者$p\cdot\alpha+2\sqrt{p\cdot\alpha+p}<n\leq \frac{\left( \alpha^{2}+1\right) \cdot p+\left( \alpha-1\right) ^{2}}{\alpha-1}$且$\frac{\alpha\left( \alpha-1\right) }{\alpha+1}\leq p$时，连通的$\mathbf{W}_{p}$图$G$的独立多项式是log-凹的。 此外，团冠图$G\circ K_{p}$是$\mathbf{W}_{p}$图类的一个例子。 我们进一步证明，对于足够大的$p$，$G\circ K_{p}$的独立多项式始终是log-凹的。 关键词：非常良好覆盖图；准正则可化图；冠图；$\mathbf{W}_{p}$图；独立多项式；log-凹性。