Depths of Powers of the Edge Ideal of a Tree

An upper bound for the regularity of powers of edge ideals

‎A recent result due to Ha and Van Tuyl proved that the Castelnuovo-Mumford regularity of the quotient ring $R/I(G)$ is at most matching number of $G$‎, ‎denoted by match$(G)$‎. ‎In this paper‎, ‎we provide a generalization of this result for powers of edge ideals‎. ‎More precisely‎, ‎we show that for every graph $G$ and every $sgeq 1$‎, ‎$${rm reg}( R‎/ ‎I(G)^{s})leq (2s-1) |E(G)|^{s-1} {rm ma...

Distinguishing number and distinguishing index of natural and fractional powers of graphs

‎The distinguishing number (resp. index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$‎ ‎such that $G$ has an vertex labeling (resp. edge labeling) with $d$ labels that is preserved only by a trivial‎ ‎automorphism‎. ‎For any $n in mathbb{N}$‎, ‎the $n$-subdivision of $G$ is a simple graph $G^{frac{1}{n}}$ which is constructed by replacing each edge of $G$ with a path of length $n$...

Linear Resolutions of Powers of Generalized Mixed Product Ideals

Let L be the generalized mixed product ideal induced by a monomial ideal I. In this paper we compute powers of the genearlized mixed product ideals and show that Lk  have a linear resolution if and only if Ik have a linear resolution for all k. We also introduce the generalized mixed polymatroidal ideals and prove that powers and monomial localizations of a generalized mixed polymatroidal ideal...

Bounds for the Regularity of Edge Ideal of Vertex Decomposable and Shellable Graphs

Topics on the Ratliff-Rush Closure of an Ideal

Introduction Let  be a Noetherian ring with unity and    be a regular ideal of , that is,  contains a nonzerodivisor. Let . Then . The :union: of this family, , is an interesting ideal first studied by Ratliff and Rush in [15]. ‎  The Ratliff-Rush closure of  ‎ is defined by‎ . ‎ A regular ideal  for which ‎‎ is called Ratliff-Rush ideal.‎‏‎ ‎ The present paper, reviews some of the known prop...