arXiv subjects

Ashkan Nikseresht

Publications and source records attributed to Ashkan Nikseresht.

15 recordsLinked to original sources

arXiv2024

Linear Complementary Equi-Dual Codes

We call a linear code $C$ with length $n$ over a field $F$, a linear complementary equi-dual code, when there exists a linear code $D$ over $F$ such that $D$ is permutation equivalent to $C^\perp$ and $(C,D)$ is a linear complementary pair of codes, that is, $C+ D=F^n$ and $C\cap D=0$. We first state a necessary condition on a code $C$ to be linear complementary equi-dual. Then, we conjecture that this necessary condition is also sufficient and present several statements which support this conjecture.

cs.IT↗

arXiv2024

Well-covered Unit Graphs of Finite Rings

Let $R$ be a finite ring with identity. The unit graph (unitary Cayley graph) of $R$ is the graph with vertex set $R$, where two distinct vertices $x$ and $y$ are adjacent exactly whenever $x+y$ is a unit in $R$ ($x-y$ is a unit in $R$). Here, we study independent sets of unit graphs of matrix rings over finite fields and use them to characterize all finite rings for which the unit graph is well-covered or Cohen-Macaulay. Moreover, we show that the unit graph of $R$ is well-covered if and only if the unitary Cayley graph of $R$ is well-covered and the characteristic of $R/J(R)$

math.CO↗

arXiv2023

Well-covered Unitary Cayley Graphs of Matrix Rings over Finite Fields and Applications

Suppose that $F$ is a finite field and $R=M_n(F)$ is the ring of $n$-square matrices over $F$. Here we characterize when the Cayley graph of the additive group of $R$ with respect to the set of invertible elements of $R$, called the unitary Cayley graph of $R$, is well-covered. Then we apply this to characterize all finite rings with identity whose unitary Cayley graph is well-covered or Cohen-Macaulay.

math.CO↗

arXiv2022

One-Sided Repeated-Root Two-Dimensional Cyclic and Constacyclic Codes

In this paper, we study some repeated-root two-dimensional cyclic and constacyclic codes over a finite field $F=\mathbb{F}_q$. We obtain the generator matrices and generator polynomials of these codes and their duals. We also investigate when such codes are self-dual. Moreover, we prove that if there exists an asymptotically good family of one-sided repeated-root two-dimensional cyclic or constacyclic codes, then there exists an asymptotically good family of simple root two-dimensional cyclic or constacyclic codes with parameters at least as good as the first family. Furthermore, we show that several of the main results of the papers Rajabi and Khashyarmanesh (2018) and Sepasdar and Khashyarmanesh (2016) are not accurate and find other conditions needed for them to hold.

cs.IT↗

arXiv2021

Gorenstein and Cohen-Macaulay Matching Complexes

Let $H$ be a simple undirected graph. The family of all matchings of $H$ forms a simplicial complex called the matching complex of $H$. Here , we give a classification of all graphs with a Gorenstein matching complex. Also we study when the matching complex of $H$ is Cohen-Macaulay and, in certain classes of graphs, we fully characterize those graphs which have a Cohen-Macaulay matching complex. In particular, we characterize when the matching complex of a graph with girth at least 5 or a complete graph is Cohen-Macaulay.

math.AC↗

arXiv2020

Some Combinatorial Characterizations of Gorenstein Graphs with Independence Number Less than Four

Let $α=α(G)$ be the independence number of a simple graph $G$ with $n$ vertices and $I(G)$ be its edge ideal in $S=K[x_1,\ldots, x_n]$. If $S/I(G)$ is Gorenstein, the graph $G$ is called Gorenstein over $K$ and if $G$ is Gorenstein over every field, then we simply say that $G$ is Gorenstein. In this article, first we state a condition equivalent to $G$ being Gorenstein and using this we give a characterization of Gorenstein graphs with $α=2$. Then we present some properties of Gorenstein graphs with $α=3$ and as an application of these results we characterize triangle-free Gorenstein graphs with $α=3$.

math.CO↗

arXiv2020

Algebraic Properties of Clique Complexes of Line Graphs

Let $H$ be a simple undirected graph and $G=\mathrm{L}(H)$ be its line graph. Assume that $Δ(G)$ denotes the clique complex of $G$. We show that $Δ(G)$ is sequentially Cohen-Macaulay if and only if it is shellable if and only if it is vertex decomposable. Moreover if $Δ(G)$ is pure, we prove that these conditions are also equivalent to being strongly connected. Furthermore, we state a complete characterizations of those $H$ for which $Δ(G)$ is Cohen-Macaulay, sequentially Cohen-Macaulay or Gorenstein. We use these characterizations to present linear time algorithms which take a graph $G$, check whether $G$ is a line graph and if yes, decide if $Δ(G)$ is Cohen-Macaulay or sequentially Cohen-Macaulay or Gorenstein.

math.AC↗

arXiv2020

On Generalizations of Cycles and Chordality to Hypergraphs from an Algebraic Viewpoint

In this paper, we study the notion of chordality and cycles in hypergraphs from a commutative algebraic point of view. The corresponding concept of chordality in commutative algebra is having a linear resolution. However, there is no unified definition for cycle or chordality in hypergraphs in the literature, so we consider several generalizations of these notions and study their algebraic interpretations. In particular, we investigate the relationship between chordality and having linear quotients in some classes of hypergraphs. Also we show that if $\mathcal{C}$ is a hypergraph such that $\langle \mathcal{C} \rangle$ is a vertex decomposable simplicial complex or $I(\bar{\mathcal{C}})$ is squarefree stable, then $\mathcal{C}$ is chordal according to one of the most promising definitions.

math.CO↗

arXiv2019

Trung's Construction and the Charney-Davis Conjecture

We consider a construction by which we obtain a simple graph $\mathrm{T}(H,v)$ from a simple graph $H$ and a non-isolated vertex $v$ of $H$. We call this construction "Trung's construction". We prove that $\mathrm{T}(H,v)$ is well-covered, W$_2$ or Gorenstein if and only if $H$ is so. Also we present a formula for computing the independence polynomial of $\mathrm{T}(H,v)$ and investigate when $\mathrm{T}(H,v)$ satisfies the Charney-Davis conjecture. As a consequence of our results, we show that every Gorenstein planar graph with girth at least four, satisfies the Charney-Davis conjecture.

math.AC↗

arXiv2019

On Gorenstein Circulant Graphs and Gorenstein SQC Graphs

We characterize some graphs with a Gorenstein edge ideal. In particular, we show that if $G$ is a circulant graph with vertex degree at most four or a circulant graph of the form $C_n(1,\ldots, d)$ for some $d\leq n/2$, then $G$ is Gorenstein if and only if $G\cong tK_2$, $G\cong t\overline{C_n}$ or $G\cong tC_{13}(1,5)$ for some integers $t$ and $n\geq 4$. Also we prove that if $G$ is a \mathcal{SQC}\ graph, then $G$ is Gorenstein if and only if each component of $G$ is either an edge or a 5-cycle.

math.AC↗

arXiv2018

Chordality of Clutters with Vertex Decomposable Dual and Ascent of Clutters

In this paper, we consider the generalization of chordal graphs to clutters proposed by Bigdeli, et al in J. Combin. Theory, Series A (2017). Assume that $\mathcal{C}$ is a $d$-dimensional uniform clutter. It is known that if $\mathcal{C}$ is chordal, then $I(\bar{\mathcal{C}})$ has a linear resolution over all fields. The converse has recently been rejected, but the following question which poses a weaker version of the converse is still open: "if $I(\bar{\mathcal{C}})$ has linear quotients, is $\mathcal{C}$ necessarily chordal?". Here, by introducing the concept of the ascent of a clutter, we split this question into two simpler questions and present some clues in support of an affirmative answer. In particular, we show that if $I(\bar{\mathcal{C}})$ is the Stanley-Reisner ideal of a simplicial complex with a vertex decomposable Alexander dual, then $\mathcal{C}$ is chordal.

math.AC↗

arXiv2017

Factorizations in Modules and Splitting Multiplicatively Closed Subsets

We introduce the concept of multiplicatively closed subsets of a commutative ring $R$ which split an $R$-module $M$ and study factorization properties of elements of $M$ with respect to such a set. Also we demonstrate how one can utilize this concept to investigate factorization properties of $R$ and deduce some Nagata type theorems relating factorization properties of $R$ to those of its localizations, when $R$ is an integral domain.

math.AC↗

arXiv2016

Finite Commutative Rings with a MacWilliams Type Relation for the m-Spotty Hamming Weight Enumerators

Let $R$ be a finite commutative ring. We prove that a MacWilliams type relation between the m-spotty weight enumerators of a linear code over $R$ and its dual hold, if and only if, $R$ is a Frobenius (equivalently, Quasi-Frobenius) ring, if and only if, the number of maximal ideals and minimal ideals of $R$ are the same, if and only if, for every linear code $C$ over $R$, the dual of the dual $C$ is $C$ itself. Also as an intermediate step, we present a new and simpler proof for the commutative case of Wood's theorem which states that $R$ has a generating character if and only if $R$ is a Frobenius ring.

math.AC↗

arXiv2016

Dual of Codes over Finite Quotients of Polynomial Rings

Let $A=\frac{\mathbb{F}[x]}{\langle f(x)\rangle }$, where $f(x)$ is a monic polynomial over a finite field $\mathbb{F}$. In this paper, we study the relation between $A$-codes and their duals. In particular, we state a counterexample and a correction to a theorem of Berger and El Amrani (Codes over finite quotients of polynomial rings, \emph{Finite Fields Appl.} \textbf{25} (2014), 165--181) and present an efficient algorithm to find a system of generators for the dual of a given $A$-code. Also we characterize self-dual $A$-codes of length 2 and investigate when the $\mathbb{F}$-dual of $A$-codes are $A$-codes.

cs.IT↗

arXiv2016

Tame graphs, clutters and their Rees algebras

A tame ideal is an ideal $I$ such that the blowup of the affine space $\mathbb{A}_k^n$ along $I$ is regular. In this paper, we give a combinatorial characterization of tame squarefree monomial ideals. More precisely, we show that a square free monomial ideal is tame if and only if the corresponding clutter is a union of some isolated vertices and a complete $d$-partite $d$-uniform clutter. It turns out that a squarefree monomial ideal is tame, if and only if the facets of its Stanley-Reisner complex have mutually disjoint complements. Also, we characterize all monomial ideals generated in degree at most 2 which are tame. Finally, we prove that tame squarefree ideals are of fiber type.

math.AC↗