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Mohammad Farrokhi Derakhshandeh Ghouchan

Publications and source records attributed to Mohammad Farrokhi Derakhshandeh Ghouchan.

At least 19 recordsLinked to original sources
arXiv2025

Positive matching decompositions of the cartesian product of graphs

Let $Γ=(V,E)$ be a finite simple graph. A matching $M \subseteq E$ is positive if there exists a weight function on $V$ such that the matching $M$ is characterized by those edges with positive weights. A positive matching decomposition (pmd) of $Γ$ with $p$ parts is an ordered partition $E_1,\ldots,E_p$ of $E$ such that $E_i$ is a positive matching of $(V, E \setminus \bigcup_{j=1}^{i-1} E_j)$, for $i = 1, \ldots, p$. The smallest $p$ for which $Γ$ admits a pmd with $p$ parts is denoted by $\mathrm{pmd}(Γ)$. We study the pmd of the Cartesian product of graphs and give sharp upper bounds for them in terms of the pmds and chromatic numbers of their components. In special cases, we compute the pmd of grid graphs that is the Cartesian product of paths and cycles.

math.CO↗
arXiv2023

Gröbner basis and Krull dimension of Lovász-Saks-Sherijver ideal associated to a tree

Let $\mathbb{K}$ be a field and $n$ be a positive integer. Let $Γ=([n], E)$ be a simple graph, where $[n]=\{1,\ldots, n\}$. If $S=\mathbb{K}[x_1, \ldots, x_n, y_1, \ldots, y_n]$ is a polynomial ring, then the graded ideal \[ L_Γ^\mathbb{K}(2) = \left( x_{i}x_{j} + y_{i}y_{j} \colon \quad \{i, j\} \in E(Γ)\right) \subset S,\] is called the Lovász-Saks-Schrijver ideal, LSS-ideal for short, of $Γ$ with respect to $\mathbb{K}$. In the present paper, we compute a Gröbner basis of this ideal with respect to lexicographic ordering induced by $x_1>\cdots>x_n>y_1>\cdots>y_n$ when $Γ=T$ is a tree. As a result, we show that it is independent of the choice of the ground field $\mathbb{K}$ and compute the Hilbert series of $L_T^\mathbb{K}(2)$. Finally, we present concrete combinatorial formulas to obtain the Krull dimension of $S/L_T^\mathbb{K}(2)$ as well as lower and upper bounds for Krull dimension.

math.AC↗
arXiv2023

Lattice paths inside a table, II

Consider an $m\times n$ table $T$ and latices paths $ν_1,\ldots,ν_k$ in $T$ such that each step $ν_{i+1}-ν_i=(1,1)$, $(1,0)$ or $(1,-1)$. The number of paths from the $(1,i)$-blank (resp. first column) to the $(s,t)$-blank is denoted by $\mathcal{D}^i(s,t)$ (resp. $\mathcal{D}(s,t)$). Also, the number of all paths form the first column to the las column is denoted by $\mathcal{I}_m(n)$. We give explicit formulas for the numbers $\mathcal{D}^1(s,t)$ and $\mathcal{D}(s,t)$.

math.GM↗
arXiv2022

Cellular resolutions of monomial ideals and their Artinian reductions

The question we address in this paper is: which monomial ideals have minimal cellular resolutions, that is, minimal resolutions obtained from homogenizing the chain maps of CW-complexes? Velasco gave families of examples of monomial ideals that do not have minimal cellular resolutions, but those examples have large minimal generating sets. In this paper, we show that if a monomial ideal has at most four generators, then the ideal and its (monomial) Artinian reductions have minimal cellular resolutions. When the ideal is generated by two monomials, we can give a precise description of the CW-complex supporting minimal free resolution of the ideal and its Artinian reduction. Also, in this case, we compute the multigraded Betti numbers, Cohen-Macaulay type and determine when the corresponding algebra is a level algebra.

math.AC↗
arXiv2022

Positive matching decompositions of graphs

A matching $M$ in a graph $Γ$ is positive if $Γ$ has a vertex-labeling such that $M$ coincides with the set of edges with positive weights. A positive matching decomposition (pmd) of $Γ$ is an edge-partition $M_1,\ldots,M_p$ of $Γ$ such that $M_i$ is a positive matching in $Γ-M_1\cup\cdots\cup M_{i-1}$, for $i=1,\ldots,p$. The pmds of graphs are used to study algebraic properties of the Lovász-Saks-Schrijver ideals arising from orthogonal representations of graphs. We give a characterization of pmds of graphs in terms of alternating closed walks and apply it to study pmds of various classes of graphs including complete multipartite graphs, (regular) bipartite graphs, cacti, generalized Petersen graphs, etc. We further show that computation of pmds of a graph can be reduced to that of its maximum pendant-free subgraph.

math.CO↗
arXiv2022

Green-Lazarsfeld index of square-free monomial ideals and their powers

Let $\mathbb{K}$ be a field and $I$ be a square-free monomial ideal in the polynomial ring $\mathbb{K}[x_1, \ldots, x_n]$. The Green-Lazarsfeld index, $\mathrm{index}(I)$, counts the number of steps to reach to a syzygy minimally generated by a nonlinear form in a graded minimal free resolution of $I$. In this paper, we study this invariant for $I$ and its powers from a combinatorial point of view. We characterize all square-free monomial ideals $I$ generated in degree $3$ such that $\mathrm{index}(I)>1$. Utilizing this result, we also characterize all square-free monomial ideals generated in degree $3$ such that $\mathrm{index}(I)>1$ and $\mathrm{index}(I^2)=1$. In case $n\leq5$, it is shown that $\mathrm{index}(I^k)>1$ for all $k$ if $I$ is any square-free monomial ideal with $\mathrm{index}(I)>1$.

math.AC↗
arXiv2022

Fully reducible simple Venn diagrams

We generalize Venn diagrams in spaces of arbitrary dimension $\geq 2$ and study simple Venn diagrams with the reducing property. Three equivalent conditions for a simple Venn diagram to reduce it completely and a classification of those diagrams is discussed. For example, we show that a simple $m$-dimensional $n$-Venn diagram is fully reducible if $n\leq m+1$ and conjecture that the converse is also true. An application of the generalized Venn diagrams and some more open problems are given.

math.CO↗
arXiv2021

Extending simplicial complexes: Topological and combinatorial properties

Given an arbitrary hypergraph $\mathcal{H}$, we may glue to $\mathcal{H}$ a family of hypergraphs to get a new hypergraph $\mathcal{H}'$ having $\mathcal{H}$ as an induced subhypergraph. In this paper, we introduce three gluing techniques for which the topological and combinatorial properties (such as Cohen-Macaulayness, shellability, vertex-decomposability etc.) of the resulting hypergraph $\mathcal{H}'$ is under control in terms of the glued components. This enables us to construct broad classes of simplicial complexes containing a given simplicial complex as induced subcomplex satisfying nice topological and combinatorial properties. Our results will be accompanied with some interesting open problems.

math.AC↗
arXiv2020

Some Algebraic Properties of Sierpiński-Type Graphs

This paper deals with some of the algebraic properties of Sierpiński graphs and a family of regular generalized Sierpiński graphs. For the family of regular generalized Sierpiński graphs, we obtain their spectrum and characterize those graphs that are Cayley graphs. As a by-product, a new family of non-Cayley vertex-transitive graphs, and consequently, a new set of non-Cayley numbers are introduced. We also obtain the Laplacian spectrum of Sierpiński graphs in some particular cases, and make a conjecture on the general case.

math.CO↗
arXiv2019

Finite groups with five relative commutativity degrees

We classify all finite groups with five relative commutativity degrees. Also, we give a partial answer to our previous conjecture on a lower bound of the number of relative commutativity degrees of finite groups.

math.GR↗
arXiv2019

Factorization graphs of finite groups

We introduce the factorization graph of a finite group and study its connectedness and forbidden structures. We characterize all finite groups with connected factorization graphs and classify those with connected bipartite factorization graphs. Also, we obtain a classification of all groups with claw-free, $K_{1,4}$-free, and square-free factorization graphs, excluding the non-solvable groups in the latter case.

math.GR↗
arXiv2019

Lattice paths inside a table: Rows and columns linear combinations

A lattice path inside the $m\times n$ table $T$ is a sequence $ν_1,\ldots,ν_k$ of cells such that $ν_{j+1}-ν_j\in\{(1,-1),(1,0),(1,1)\}$ for all $j=1,\ldots,k-1$. The number of lattice paths in $T$ from the first column to the $(x,y)$-cell is written into that cell. We present a precise description of the minimal linear recurrences among rows, columns, and columns sums. As a result, we obtain several formulas for the number of all lattice paths from the first column to the last column of $T$, that is, the $n^{th}$ column sum. Our methods are based on three classes of operators, which will also be studied independently.

math.CO↗
arXiv2019

Lattice paths inside a table, I

A lattice path in $\mathbb{Z}^d$ is a sequence $ν_1,ν_2,\ldots,ν_k\in\mathbb{Z}^d$ such that the steps $ν_i-ν_{i-1}$ lie in a subset $\mathbf{S}$ of $\mathbb{Z}^d$ for all $i=2,\ldots,k$. Let $T_{m,n}$ be the $m\times n$ table in the first area of the $xy$-axis and put $\mathbf{S}=\{(1,1),(1,0),(1,-1)\}$. Accordingly, let $\mathcal{I}_m(n)$ denote the number of lattice paths starting from the first column and ending at the last column of $T$. We will study the numbers $\mathcal{I}_m(n)$ and give explicit formulas for special values of $m$ and $n$. As a result, we prove a conjecture of \textit{Alexander R. Povolotsky} involving $\mathcal{I}_n(n)$. Finally, we present some relationships between the number of lattice paths and Fibonacci and Pell-Lucas numbers, and pose an open problem.

math.CO↗
arXiv2016

A characterization of Tutte-Coxeter graph

We give a natural generalization of the Tutte-Coxeter graph in a natural way and prove that the Tutte-Coxeter graph is the only vertex-transitive (edge-transitive) graph among all generalized Tutte-Coxeter graphs.

math.CO↗
arXiv2015

Relative Cayley graphs of finite groups

The relative Cayley graph of a group $G$ with respect to its proper subgroup $H$, is a graph whose vertices are elements of $G$ and two vertices $h\in H$ and $g\in G$ are adjacent if $g=hc$ for some $c\in C$, where $C$ is an inversed-closed subset of $G$. We study the relative Cayley graphs and, among other results, we discuss on their connectivity and forbidden structures, and compute some of their important numerical invariants.

math.CO↗
arXiv2015

Self 2-distance graphs

All finite simple self $2$-distance graphs with no $4$-cycle, diamond, or triangles with a common vertex are determined. Utilizing these results, it is shown that there is no cubic self $2$-distance graphs.

math.CO↗