Equivalence of Extensions and the Perturbation Index
In this paper, using the equivalence of two extensions, we obtain an improved upper bound for the perturbation index, improving the bounds of Quy and N. V. Trung and of Tan.
math.AC↗
arXiv subjects
Ton That Quoc Tan
Publications and source records attributed to Ton That Quoc Tan.
An Improved Upper Bound for the Perturbation Index
Let $(R,\mathfrak m)$ be a Noetherian local ring and $J$ be an arbitrary ideal of $R$. Suppose that $f_1,\ldots,f_r$ is a $J$-filter regular sequence in $R$ and $I=(f_1,\ldots,f_r)$. In this paper, we establish an improved upper bound for the perturbation index of the associated graded ring $\mathrm{gr}_J(R/I)$, refining the bound obtained by Quy and N. V. Trung.
math.AC↗
Independence Polynomials of graphs and degree of $h$-polynomials of edge ideals
Let $G=(V,E)$ be a finite simple graph. In this paper, we study the degree of the $h$-polynomial of the edge ideal of $G$ in relation to the independence number of $G$. Our approach is based on the value of the independence polynomial of $G$ at $-1$ and its derivatives at $-1$. We establish a necessary and sufficient condition for the equality $° h_{R/I(G)}(t)=α(G)$. As consequences, we obtain combinatorial formulas for the degree of the $h$-polynomial for several classes of graphs, including paths, cycles, bipartite graphs, Cameron--Walker graphs, and antiregular graphs.
math.AC↗