arXiv subjects
Publications and source records attributed to Shiya Duan.
This paper gives exact formulas for the regularity of edge ideals of edge-weighted integrally closed trees. In addition, we provide some linear upper bounds on the regularity of powers of such ideals.
This paper presents exact formulas for the regularity and depth of powers of edge ideals of an edge-weighted star graph. Additionally, we provide exact formulas for the regularity of powers of the edge ideal of an edge-weighted integrally closed path, as well as lower bounds on the depth of powers of such an edge ideal.
Let $G$ be a finite simple graph with the vertex set $V$ and let $I_G$ be its edge ideal in the polynomial ring $S=\mathbb{K}[x_V]$. In this paper, we compute the depth and the Castelnuovo--Mumford regularity of $S/I_G$ when $G=G_1\circ G_2$ or $G=G_1* G_2$ is a graph obtained from Cohen-Macaulay bipartite graphs $G_1$, $G_2$ by $\circ$ operation or $*$ operation, respectively.
Let $G_ω$ be an edge-weighted simple graph. In this paper, we give a complete characterization of the graph $G_ω$ whose edge ideal $I(G_ω)$ is integrally closed. We also show that if $G_ω$ is an edge-weighted star graph, a path or a cycle, and $I(G_ω)$ is integrally closed, then $I(G_ω)$ is normal.
An equigenerated monomial ideal $I$ in the polynomial ring $S= K[x_1,\ldots,x_n]$ is a Freiman ideal if $μ(I^2)=\ell(I)μ(I)-{\ell(I)\choose 2}$ where $\ell(I)$ is the analytic spread of $I$ and $μ(I)$ is the number of minimal generators of $I$. In this paper, we classify certain classes of Borel type ideals, including Borel ideals with multiple Borel generators and principal $k$-Borel ideals, which are Freiman.