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Let $G$ be a finite simple graph on the vertex set $V(G) = {x_1, ldots, x_n}$ and $I(G) subset K[V(G)]$ its edge ideal, where $K[V(G)]$ is the polynomial ring in $x_1, ldots, x_n$ over a field $K$ with each ${rm deg} x_i = 1$ and where $I(G)$ is generated by those squarefree quadratic monomials $x_ix_j$ for which ${x_i, x_j}$ is an edge of $G$. In the present paper, given integers $1 leq a leq r$ and $s geq 1$, the existence of a finite connected simple graph $G = G(a, r, d)$ with ${rm im}(G) = a$, ${rm reg}(R/I(G)) = r$ and ${rm deg} h_{K[V(G)]/I(G)} (lambda) = s$, where ${rm im}(G)$ is the induced matching number of $G$ and where $h_{K[V(G)]/I(G)} (lambda)$ is the $h$-polynomial of $K[V(G)]/I(G)$.
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We study homological properties of random quadratic monomial ideals in a polynomial ring $R = {mathbb K}[x_1, dots x_n]$, utilizing methods from the Erd{o}s-R{e}nyi model of random graphs. Here for a graph $G sim G(n, p)$ we consider the `coedge ideal $I_G$ corresponding to the missing edges of $G$, and study Betti numbers of $R/I_G$ as $n$ tends to infinity. Our main results involve fixing the edge probability $p = p(n)$ so that asymptotically almost surely the Krull dimension of $R/I_G$ is fixed. Under these conditions we establish various properties regarding the Betti table of $R/I_G$, including sharp bounds on regularity and projective dimension, and distribution of nonzero normalized Betti numbers. These results extend work of Erman and Yang, who studied such ideals in the context of conjectured phenomena in the nonvanishing of asymptotic syzygies. Along the way we establish results regarding subcomplexes of random clique complexes as well as notions of higher-dimensional vertex $k$-connectivity that may be of independent interest.
Let $L_n$ be a line graph with $n$ edges and $F(L_n)$ the facet ideal of its matching complex. In this paper, we provide the irreducible decomposition of $F(L_n)$ and some exact formulas for the projective dimension and the regularity of $F(L_n)$.
In this paper we prove the conjectured upper bound for Castelnuovo-Mumford regularity of binomial edge ideals posed in [23], in the case of chordal graphs. Indeed, we show that the regularity of any chordal graph G is bounded above by the number of maximal cliques of G, denoted by c(G). Moreover, we classify all chordal graphs G for which L(G) = c(G), where L(G) is the sum of the lengths of longest induced paths of connected components of G. We call such graphs strongly interval graphs. Moreover, we show that the regularity of a strongly interval graph G coincides with L(G) as well as c(G).
We characterize unmixed and Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs. We also provide examples of oriented graphs which have unmixed and non-Cohen-Macaulay vertex-weighted edge ideals, while the edge ideal of their underlying graph is Cohen-Macaulay. This disproves a conjecture posed by Pitones, Reyes and Toledo.
Let $mathcal{D}$ be a weighted oriented graph and $I(mathcal{D})$ be its edge ideal. In this paper, we show that all the symbolic and ordinary powers of $I(mathcal{D})$ coincide when $mathcal{D}$ is a weighted oriented certain class of tree. Finally, we give necessary and sufficient conditions for the equality of ordinary and symbolic powers of naturally oriented lines.
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