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Nonplanarity of unit graphs and classification of the toroidal ones
, Article Pacific Journal of Mathematics ; Vol. 268, Issue. 2 , 2014 , pp. 371-387 ; ISSN: 0030-8730 ; Maimani, H. R ; Pournaki, M. R ; Yassemi, S ; Sharif University of Technology
Abstract
The unit graph of a ring R with nonzero identity is the graph in which the vertex set is R, and two distinct vertices x and y are adjacent if and only if x + y is a unit in R. In this paper, we derive several necessary conditions for the nonplanarity of the unit graphs of finite commutative rings with nonzero identity, and determine, up to isomorphism, all finite commutative rings with nonzero identity whose unit graphs are toroidal
An ideal theoretic approach to complete partite zero-divisor graphs of posets
, Article Journal of Algebra and its Applications ; Volume 12, Issue 2 , 2013 ; 02194988 (ISSN) ; Maimani, H. R ; Pournaki, M. R ; Yassemi, S ; Sharif University of Technology
2013
Abstract
In this paper, we characterize complete partite zero-divisor graphs of posets via the ideals of the posets. In particular, for complete bipartite zero-divisor graphs, we give a characterization based on the prime ideals of the posets
An existence-uniqueness theorem for a class of boundary value problems
, Article Fixed Point Theory ; Volume 13, Issue 2 , 2012 , Pages 589-592 ; 15835022 (ISSN) ; Pournaki, M. R ; Razani, A ; Sharif University of Technology
2012
Abstract
In this paper the solutions of a two-endpoint boundary value problem is studied and under suitable assumptions the existence and uniqueness of a solution is proved. As a consequence, a condition to guarantee the existence of at least one periodic solution for a class of Liénard equations is presented
On the diameter and girth of zero-divisor graphs of posets
, Article Discrete Applied Mathematics ; Volume 160, Issue 9 , 2012 , Pages 1319-1324 ; 0166218X (ISSN) ; Das, A. K ; Maimani, H. R ; Pournaki, M. R ; Yassemi, S ; Sharif University of Technology
2012
Abstract
In this paper, we deal with zero-divisor graphs of posets. We prove that the diameter of such a graph is either 1, 2 or 3 while its girth is either 3, 4 or ∞. We also characterize zero-divisor graphs of posets in terms of their diameter and girth
On the h-vector of a simplicial complex with Serre's condition
, Article Journal of Pure and Applied Algebra ; Volume 216, Issue 1 , January , 2012 , Pages 91-94 ; 00224049 (ISSN) ; Pournaki, M. R ; Seyed Fakhari, S. A ; Yassemi, S
2012
Abstract
Let δ be a (d-1)-dimensional simplicial complex and let h(δ)=(h0,h1,...,hd) be its h-vector. A recent result of Murai and Terai guarantees that if δ satisfies Serre's condition (Sr), then (h0,h1,...,hr) is an M-vector and hr+hr+1+...+hd is nonnegative. In this article, we extend the result of Murai and Terai by giving r extra necessary conditions. More precisely, we prove that if δ satisfies Serre's condition (Sr), then iihr+i+1ihr+1+...+i+d-rihd, 0≤i≤r≤d, are all nonnegative
Graphs attached to rings revisited
, Article Arabian Journal for Science and Engineering ; Volume 36, Issue 6 , 2011 , Pages 997-1011 ; 13198025 (ISSN) ; Pournaki, M. R ; Tehranian, A ; Yassemi, S ; Sharif University of Technology
Abstract
In this paper, we discuss some recent results on graphs attached to rings. In particular, we deal with comaximal graphs, unit graphs, and total graphs. We then define the notion of cototal graph and, using this graph, we characterize the rings which are additively generated by their zero divisors. Finally, we glance at graphs attached to other algebraic structures
Necessary and sufficient conditions for unit graphs to be Hamiltonian
, Article Pacific Journal of Mathematics ; Volume 249, Issue 2 , February , 2011 , Pages 419-429 ; 00308730 (ISSN) ; Pournaki, M. R ; Yassemi, S ; Sharif University of Technology
2011
Abstract
The unit graph corresponding to an associative ring R is the graph obtained by setting all the elements of R to be the vertices and defining distinct vertices x and y to be adjacent if and only if x + y is a unit of R. By a constructive method, we derive necessary and sufficient conditions for unit graphs to be Hamiltonian
Classification of rings with unit graphs having domination number less than four
, Article Rendiconti del Seminario Matematico dell 'Universita' di Padova/Mathematical Journal of the University of Padova ; Volume 133 , 2015 , Pages 173-195 ; 00418994 (ISSN) ; Maimani, H. R ; Pournaki, M. R ; Yassemi, S ; Sharif University of Technology
Universita di Padova
2015
Abstract
Let R be a finite commutative ring with nonzero identity. The unit graph of R is the graph obtained by setting all the elements of R to be the vertices and defining distinct vertices x and y to be adjacent if and only if x + y is a unit ele¬ment of R. In this paper, a classification of finite commutative rings with nonzero identity in which their unit graphs have domination number less than four is given
Cohen-Macaulayness and Limit Behavior of Depth for Powers of Cover Ideals
, Article Communications in Algebra ; Volume 43, Issue 1 , Aug , 2015 , Pages 143-157 ; 00927872 (ISSN) ; Pournaki, M. R ; Seyed Fakhari, S. A ; Terai, N ; Yassemi, S ; Sharif University of Technology
Taylor and Francis Inc
2015
Abstract
Let K{double-struck} be a field, and let R = K{double-struck}[x1,.., xn] be the polynomial ring over K{double-struck} in n indeterminates x1,.., xn. Let G be a graph with vertex-set {x1,.., xn}, and let J be the cover ideal of G in R. For a given positive integer k, we denote the kth symbolic power and the kth bracket power of J by J(k) and J[k], respectively. In this paper, we give necessary and sufficient conditions for R/Jk, R/J (k), and R/J [k] to be Cohen-Macaulay. We also study the limit behavior of the depths of these rings
A Class of Weakly Perfect Graphs
, Article Czechoslovak Mathematical Journal ; Volume 60, Issue 4 , 2010 , Pages 1037-1041 ; 00114642 (ISSN) ; Pournaki, M. R ; Yassemi, S ; Sharif University of Technology
Abstract
A graph is called weakly perfect if its chromatic number equals its clique number. In this note a new class of weakly perfect graphs is presented and an explicit formula for the chromatic number of such graphs is given
A note on periodic solutions of Riccati equations
, Article Nonlinear Dynamics ; Volume 62, Issue 1-2 , 2010 , Pages 119-125 ; 0924090X (ISSN) ; Pournaki, M. R ; Razani, A ; Sharif University of Technology
2010
Abstract
In this note, we show that under certain assumptions the scalar Riccati differential equation x′=a(t)x+b(t)x 2+c(t) with periodic coefficients admits at least one periodic solution. Also, we give two illustrative examples in order to indicate the validity of the assumptions
Weakly perfect graphs arising from rings
, Article Glasgow Mathematical Journal ; Volume 52, Issue 3 , 2010 , Pages 417-425 ; 00170895 (ISSN) ; Pournaki, M. R ; Yassemi, S ; Sharif University of Technology
2010
Abstract
A graph is called weakly perfect if its chromatic number equals its clique number. In this paper a new class of weakly perfect graphs arising from rings are presented and an explicit formula for the chromatic number of such graphs is given. Copyright
Unit graphs associated with rings
, Article Communications in Algebra ; Volume 38, Issue 8 , 2010 , Pages 2851-2871 ; 00927872 (ISSN) ; Maimani, H. R ; Pournaki, M. R ; Yassemi, S ; Sharif University of Technology
2010
Abstract
Let R be a ring with nonzero identity. The unit graph of R, denoted by G(R), has its set of vertices equal to the set of all elements of R; distinct vertices x and y are adjacent if and only if x + y is a unit of R. In this article, the basic properties of G(R) are investigated and some characterization results regarding connectedness, chromatic index, diameter, girth, and planarity of G(R)are given. (These terms are defined in Definitions and Remarks 4.1, 5.1, 5.3, 5.9, and 5.13.)
The nonorientable genus of some Jacobson graphs and classification of the projective ones
, Article Publicationes Mathematicae ; Volume 88, Issue 3-4 , 2016 , Pages 425-437 ; 00333883 (ISSN) ; Maimani, H. R ; Pournaki, M. R ; Zaeembashi, A ; Sharif University of Technology
Kossuth Lajos Tudomanyegyetem
2016
Abstract
Let R be a finite commutative ring with nonzero identity and denote its Jacobson radical by J(R). The Jacobson graph of R is the graph in which the vertex set is RJ(R), and two distinct vertices x and y are adjacent if and only if 1-xy is not a unit in R. In this paper, the nonorientable genus of some Jacobson graphs is either computed or estimated by a lower bound. As an application, the rings R with projective Jacobson graphs are classified
Combinatorics comes to the rescue: h-vectors in commutative algebra
, Article Mathematical Intelligencer ; 2018 ; 03436993 (ISSN) ; Pournaki, M.R ; Seyed Fakhari, S.A ; Yassemi, S ; Sharif University of Technology
Springer New York LLC
2018
Classification of the toroidal jacobson graphs
, Article Bulletin of the Malaysian Mathematical Sciences Society ; Volume 41, Issue 1 , 2018 , Pages 321-334 ; 01266705 (ISSN) ; Maimani, H. R ; Pournaki, M. R ; Zaeembashi, A ; Sharif University of Technology
Springer Singapore
2018
Abstract
Let R be a finite commutative ring with nonzero identity and denote its Jacobson radical by J(R). The Jacobson graph of R is the graph in which the vertex set is R J(R) , and two distinct vertices x and y are adjacent if and only if 1 - xy is not a unit in R. In this paper, up to isomorphism, we classify the rings R whose Jacobson graphs are toroidal. © 2016, Malaysian Mathematical Sciences Society and Universiti Sains Malaysia
Combinatorics comes to the rescue: H-vectors in commutative algebra
, Article Mathematical Intelligencer ; Volume 41, Issue 1 , 2019 , Pages 16-21 ; 03436993 (ISSN) ; Pournaki, M. R ; Seyed Fakhari, S. A ; Yassemi, S ; Sharif University of Technology
Springer New York LLC
2019
Cohen–Macaulayness of two classes of circulant graphs
, Article Journal of Algebraic Combinatorics ; 2020 ; Maimani, H. R ; Mousivand, A ; Pournaki, M. R ; Sharif University of Technology
Springer
2020
Abstract
Let n be a positive integer and let Sn be the set of all nonnegative integers less than n which are relatively prime to n. In this paper, we discuss structural properties of circulant graphs generated by the Sn′s and their complements. In particular, we characterize when these graphs are well-covered, Cohen–Macaulay, Buchsbaum or Gorenstein. © 2020, Springer Science+Business Media, LLC, part of Springer Nature
Robability that an element of a finite group has a square root
, Article Colloquium Mathematicum ; Volume 112, Issue 1 , 2008 , Pages 147-155 ; 00101354 (ISSN) ; Pournaki, M. R ; Sharif University of Technology
Instytut Matematyczny
2008
Abstract
Let G be a finite group of even order. We give some bounds for the probability p(G) that a randomly chosen element in G has a square root. In particular, we prove that p(G) ≤ 1 - |√|G|/|G|. Moreover, we show that if the Sylow 2-subgroup of G is not a proper normal elementary abelian subgroup of G, then p(G) ≤ 1 - 1/√|G|. Both of these bounds are best possible upper bounds for p(G), depending only on the order of G. © Instytut Matematyczny PAN, 2008
Cohen–macaulayness of a class of graphs versus the class of their complements
, Article Discrete Mathematics ; Volume 344, Issue 10 , 2021 ; 0012365X (ISSN) ; Asir, T ; Hoang, D. T ; Pournaki, M. R ; Sharif University of Technology
Elsevier B.V
2021
Abstract
Let n≥2 be an integer. The graph G(n) is obtained by letting all the elements of {0,…,n−1} to be the vertices and defining distinct vertices x and y to be adjacent if and only if gcd(x+y,n)=1. In this paper, well-coveredness, Cohen–Macaulayness, vertex-decomposability and Gorensteinness of these graphs and their complements are characterized. These characterizations provide large classes of Cohen–Macaulay and non Cohen–Macaulay graphs. © 2021 Elsevier B.V