Log-concavity of the independence polynomials of $\mathbf{W}_{p}$ graphs
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arXiv
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| Formato: | Preprint |
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2024
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| _version_ | 1866915475781320704 |
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| author | Hoang, Do Trong Levit, Vadim E. Mandrescu, Eugen Pham, My Hanh |
| author_facet | Hoang, Do Trong Levit, Vadim E. Mandrescu, Eugen Pham, My Hanh |
| contents | 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 $n\geq(p+1)\cdotα$, where $α$ 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 $(p+1)\cdotα\leq n\leq p\cdotα+2\sqrt{p\cdotα+p}$ and $\frac{α^{2}}{4\left( α+1\right) }\leq p$, or $p\cdotα+2\sqrt{p\cdotα+p}<n\leq \frac{\left( α^{2}+1\right) \cdot p+\left( α-1\right) ^{2}}{α-1}$ and $\frac{α\left( α-1\right) }{α+1}\leq p$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00827 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Log-concavity of the independence polynomials of $\mathbf{W}_{p}$ graphs Hoang, Do Trong Levit, Vadim E. Mandrescu, Eugen Pham, My Hanh Combinatorics Discrete Mathematics 05C31, 05C69 (Primary) 05C05, 05C48 (Secondary) G.2.1; G.2.2 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 $n\geq(p+1)\cdotα$, where $α$ 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 $(p+1)\cdotα\leq n\leq p\cdotα+2\sqrt{p\cdotα+p}$ and $\frac{α^{2}}{4\left( α+1\right) }\leq p$, or $p\cdotα+2\sqrt{p\cdotα+p}<n\leq \frac{\left( α^{2}+1\right) \cdot p+\left( α-1\right) ^{2}}{α-1}$ and $\frac{α\left( α-1\right) }{α+1}\leq p$. 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. |
| title | Log-concavity of the independence polynomials of $\mathbf{W}_{p}$ graphs |
| topic | Combinatorics Discrete Mathematics 05C31, 05C69 (Primary) 05C05, 05C48 (Secondary) G.2.1; G.2.2 |
| url | https://arxiv.org/abs/2409.00827 |