arXiv subjects
Publications and source records attributed to My Hanh Pham.
We show that the independence complex of a tree is contractible if and only if it can be reduced to a path \( P_n \) with \( n \equiv 1 \pmod{3} \) by a sequence of truncation moves at branching points. As a consequence of our method, we also characterize the trees for which the independence polynomial evaluated at \( -1 \) is equal to \( 1 \) or \( -1 \).
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.
We prove that the regularity of edge ideals of powers of forests is weakly decreasing. We then compute the regularity of edge ideals of powers of cycles.
We compute the regularity of powers and symbolic powers of edge ideals of all cubic circulant graphs. In particular, we establish Conjecture of Minh for cubic circulant graphs.