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#### Zero-divisor graphs of non-commutative rings

, Article Journal of Algebra ; Volume 296, Issue 2 , 2006 , Pages 462-479 ; 00218693 (ISSN) ; Mohammadian, A ; Sharif University of Technology
2006

Abstract

In a manner analogous to the commutative case, the zero-divisor graph of a non-commutative ring R can be defined as the directed graph Γ (R) that its vertices are all non-zero zero-divisors of R in which for any two distinct vertices x and y, x → y is an edge if and only if xy = 0. We investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of Γ (R). In this paper it is shown that, with finitely many exceptions, if R is a ring and S is a finite semisimple ring which is not a field and Γ (R) ≃ Γ (S), then R ≃ S. For any finite field F and each integer n ≥ 2, we prove that if R is a ring and Γ (R) ≃ Γ (Mn), then R ≃ Mnn. Redmond defined the simple...

#### Weight Choosability of Graphs

, M.Sc. Thesis Sharif University of Technology ; Akbari, Saeed (Supervisor)
Abstract

Suppose the edges of a graph G are assigned 3-element lists of real weights. Is it possible to choose a weight for each edge from its list so that the sums of weights around adjacent vertices were different?

A graph G = (V;E) is called (k; k′)-total weight choosable if the following holds: For any total list assignment L which assigns to each vertex x a set L(x) of k real numbers, and assigns to each edge e a set L(e) of k′ real numbers, there is a mapping f : V [ E ! R such that f(y) 2 L(y) for any y 2 V [ E and for any two adjacent vertices x; x′

Σ e2E(x) f(e) + f(x) ̸=Σ e2E(x′) f(e) + f(x′)

Is it possible every graph is (2, 2)-total weight choosable and every...

A graph G = (V;E) is called (k; k′)-total weight choosable if the following holds: For any total list assignment L which assigns to each vertex x a set L(x) of k real numbers, and assigns to each edge e a set L(e) of k′ real numbers, there is a mapping f : V [ E ! R such that f(y) 2 L(y) for any y 2 V [ E and for any two adjacent vertices x; x′

Σ e2E(x) f(e) + f(x) ̸=Σ e2E(x′) f(e) + f(x′)

Is it possible every graph is (2, 2)-total weight choosable and every...

#### Orientations of graphs avoiding given lists on out-degrees

, Article Journal of Graph Theory ; Volume 93, Issue 4 , 2020 , Pages 483-502 ; Dalirrooyfard, M ; Ehsani, K ; Ozeki, K ; Sherkati, R ; Sharif University of Technology
Wiley-Liss Inc
2020

Abstract

Let G be a graph and F: V(G) → 2N be a mapping. The graph G is said to be F- avoiding if there exists an orientation O of G such that do +(v)∉ F (v) for every v ∈ V (G), where do +(v) denotes the out-degree of v in the directed graph G with respect to O. In this paper it is shown that if G is bipartite and |F (v)| ≤ dG (v)/2 for every v ∈ V (G), then G is F-avoiding. The bound |F (v)| ≤ dG (v)/2 is best possible. For every graph G, we conjecture that if |F (v)| ≤ 1/2 dG (v)-1) for every v ∈ V (G), then G is F-avoiding. We also argue about this conjecture for the best possibility of the conditions and also show some partial solutions. © 2019 Wiley Periodicals, Inc

#### Discovering associations among technologies using neural networks for tech-mining

, Article IEEE Transactions on Engineering Management ; 2020 ; Veisi, H ; Fateh rad, M ; Rahmani, R ; Sharif University of Technology
Institute of Electrical and Electronics Engineers Inc
2020

Abstract

In both public and private sectors, critical technology-based tasks, such as innovation, forecasting, and road-mapping, are faced with unmanageable complexity due to the ever-expanding web of technologies which can range into thousands. This context cannot be easily handled manually or with efficient speed. However, more precise and insightful answers are expected. These answers are the fundamental challenge addressed by tech-mining. For tech-mining, discovering the associations among them is a critical task. These associations are used to form a weighted directed graph of technologies called “association tech-graph” which is used for technology development, trend analysis, policymaking,...

#### Discovering associations among technologies using neural networks for tech-mining

, Article IEEE Transactions on Engineering Management ; Volume 69, Issue 4 , 2022 , Pages 1394-1404 ; 00189391 (ISSN) ; Veisi, H ; Fateh-Rad, M ; Rahmani, R ; Sharif University of Technology
Institute of Electrical and Electronics Engineers Inc
2022

Abstract

In both public and private sectors, critical technology-based tasks, such as innovation, forecasting, and road-mapping, are faced with unmanageable complexity due to the ever-expanding web of technologies which can range into thousands. This context cannot be easily handled manually or with efficient speed. However, more precise and insightful answers are expected. These answers are the fundamental challenge addressed by tech-mining. For tech-mining, discovering the associations among them is a critical task. These associations are used to form a weighted directed graph of technologies called 'association tech-graph' which is used for technology development, trend analysis, policymaking,...

#### Subgraphs Associated with Lists in Graphs and Digraphs

, M.Sc. Thesis Sharif University of Technology ; Akbari, Saeed (Supervisor)
Abstract

Let G be a graph and F : V (G) ...! 2N be a function. The graph G is said to be F-avoiding if there exists an orientation O of G such that d+ O(v) =2 F(v) for every v 2 V (G), where d+O(v) denotes the out-degree of v in the directed graph G with respect to O. In this thesis it is shown that if G is bipartite and jF(v)j dG(v) 2 for every v 2 V (G), then G is F-avoiding. The bound jF(v)j dG(v) 2 is best possible. For every graph G, we conjecture that if jF(v)j dG(v) 1 2 for every v 2 V (G), then G is F-avoiding. We also argue this conjecture; the best possibility and some partial solutions, e.g. for the complete graphs. For graph G, we say the set f(d+ i ; d i ); i = 1; : : : ; ng of...

#### On properties of a particular class of directed graphs used in stability analysis of flocking algorithms

, Article Proceedings of the IEEE International Conference on Control Applications, 3 October 2012 through 5 October 2012 ; 2012 , Pages 605-608 ; 1085-1992 (ISSN) ; 9781467345033 (ISBN) ; Haeri, M ; Sharif University of Technology
2012

Abstract

In this paper, we present sufficient conditions to address a larger class of digraphs, including balanced ones, whose members' Laplacian (L) makes L 1L + LTL1 to be positive semi-definite, where L1 is the Laplacian associated with a fully connected equally-edged weighted graphs. This property can be later utilized to introduce an appropriate energy function for stability analysis of flocking algorithms in a larger class of networks with switching directed information flow. Also, some of their properties are investigated in the line of matrix theory and graph theory

#### Mathematical Modeling of Diabetic Nephropathy

, M.Sc. Thesis Sharif University of Technology ; Abd Khodaei, Mohammad Jafar (Supervisor) ; Bozorgmehri, Ramin (Co-Advisor)
Abstract

Diabetes has become the most prevalent disease all among the world. One of the complications of this disease is diabetic nephropathy. Diabetic nephropathy occurs when the kidney cells encounter high glucose concentration for long periods. Kidney cells use glucose as their energy supply. However, the excess glucose in the diabetics triggers specific gene pathways, gradually leading to diabetic nephropathy. As a result, proteins’ levels and the activity of transcription factors change during the disease progression. Diabetic nephropathy is characterized by excessive deposition of the proteins comprising the extracellular matrix in the mesangium, basement membrane of the glomerulus, and in the...

#### Imprimitivity index of the adjacency matrix of digraphs

, Article Linear Algebra and Its Applications ; Volume 517 , 2017 , Pages 1-10 ; 00243795 (ISSN) ; Ghodrati, A. H ; Hosseinzadeh, M. A ; Sharif University of Technology
Elsevier Inc
2017

Abstract

Let G be a graph. An edge orientation of G is called smooth if the in-degree and the out-degree of every vertex differ by at most one. In this paper, we show that if G is a 2-edge-connected non-bipartite graph with δ(G)≥3, then G has a smooth primitive orientation. Among other results, using the spectral radius of digraphs, we show that if D1 is a primitive regular orientation and D2 is a non-regular orientation of a given graph, then for sufficiently large t, the number of closed walks of length t in D1 is more than the number of closed walks of length t in D2. © 2016 Elsevier Inc

#### Mixed paths and cycles determined by their spectrum

, Article Linear Algebra and Its Applications ; Volume 586 , 2020 , Pages 325-346 ; Ghafari, A ; Nahvi, M ; Nematollahi, M. A ; Sharif University of Technology
Elsevier Inc
2020

Abstract

A mixed graph is obtained from a graph by orienting some of its edges. The Hermitian adjacency matrix of a mixed graph with the vertex set {v1,…,vn}, is the matrix H=[hij]n×n, where hij=−hji=i if there is a directed edge from vi to vj, hij=1 if there exists an undirected edge between vi and vj, and hij=0 otherwise. The Hermitian spectrum of a mixed graph is defined to be the spectrum of its Hermitian adjacency matrix. In this paper we study mixed graphs which are determined by their Hermitian spectrum (DHS). First, we show that each mixed cycle is switching equivalent to either a mixed cycle with no directed edges (Cn), a mixed cycle with exactly one directed edge (Cn 1), or a mixed cycle...

#### Hypoenergetic and nonhypoenergetic digraphs

, Article Linear Algebra and Its Applications ; Volume 618 , 2021 , Pages 129-143 ; 00243795 (ISSN) ; Das, K. C ; Khalashi Ghezelahmad, S ; Koorepazan Moftakhar, F ; Sharif University of Technology
Elsevier Inc
2021

Abstract

The energy of a graph G, E(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. This concept was extended by Nikiforov to arbitrary complex matrices. Recall that the trace norm of a digraph D is defined as, N(D)=∑i=1nσi, where σ1≥⋯≥σn are the singular values of the adjacency matrix of D. In this paper we would like to present some lower and upper bounds for N(D). For any digraph D it is proved that N(D)≥rank(D) and the equality holds if and only if D is a disjoint union of directed cycles and directed paths. Finally, we present a lower bound on σ1 and N(D) in terms of the size of digraph D. © 2021 Elsevier Inc

#### Hypoenergetic and nonhypoenergetic digraphs

, Article Linear Algebra and Its Applications ; Volume 618 , 2021 , Pages 129-143 ; 00243795 (ISSN) ; Das, K. C ; Khalashi Ghezelahmad, S ; Koorepazan Moftakhar, F ; Sharif University of Technology
Elsevier Inc
2021

Abstract

The energy of a graph G, E(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. This concept was extended by Nikiforov to arbitrary complex matrices. Recall that the trace norm of a digraph D is defined as, N(D)=∑i=1nσi, where σ1≥⋯≥σn are the singular values of the adjacency matrix of D. In this paper we would like to present some lower and upper bounds for N(D). For any digraph D it is proved that N(D)≥rank(D) and the equality holds if and only if D is a disjoint union of directed cycles and directed paths. Finally, we present a lower bound on σ1 and N(D) in terms of the size of digraph D. © 2021 Elsevier Inc

#### Development of an Intelligent Decision Support System to Operate Gas Transmission Network

, M.Sc. Thesis Sharif University of Technology ; Bozorgmehry Boozarjomehry, Ramin (Supervisor)
Abstract

In plant-wide systems, faults can easily propagate due to the interconnection of streams. The main objective of this paper is to develop a systematic framework for online decision support system (DSS) in order to make the right decision to get the process out of critical conditions (which cannot be handled by the plant controllers) smoothly. One of the key features of the proposed approach is its lack of dependence on prior knowledge of the fault signals (e.g. number of faults, and their origin). In the proposed scheme, the plant is modeled by a fuzzy directed graph (FDG). The proposed scheme utilizes a backward reasoning based on the deviations that exists in the process variables...

#### Spectral Theory of Signed Graphs and Digraphs

, Ph.D. Dissertation Sharif University of Technology ; Akbari, Saieed (Supervisor)
Abstract

A signed graph is a pair like (G; ), where G is the underlying graph and : E(G) ! f1; +1g is a sign function on the edges of G. Here, we study the spectral determination problem for signed n-cycles (Cn; ) with respect to the adjacency spectrum and the Laplacian spectrum. In particular we prove that signed odd cycles and unbalanced even cycles are uniquely determined by their Laplacian spectrums, but balanced even cycles are not, and we find all L-cospectral mates for them. Moreover, signed odd cycles are uniquely determined by their spectrums but the signed even cycles, (except (C4;) and (C4; +)), are not and we find almost all cospectral mates for them. A mixed graph is obtained from a...

#### On the Design of Robust Multi-secret Sharing Schemes

, M.Sc. Thesis Sharif University of Technology ; Aref, Mohammad Reza (Supervisor) ; Eghlidos, Taraneh (Co-Supervisor)
Abstract

A threshold secret sharing scheme is a method for sharing a secret among a group of players, called participants. Each participant is given a private value, as his/her share. The secret can only be recovered when a certain number of participants pool their shares together. The correct recovery of the secret in these schemes depends on the honest behavior of the participants. However, in the real world, this assumption may not hold well. Dishonest participants could submit fraudulent shares in the process of recovering the secret so that honest participants cannot obtain the proper secret. A robust secret sharing scheme still makes it possible to recover the secret correctly, despite the...

#### The 2D digraph-based NoCs: Attractive alternatives to the 2D mesh NoCs

, Article Journal of Supercomputing ; Vol. 59, issue. 1 , January , 2012 , pp. 1-21 ; ISSN: 9208542 ; Modarressi, M ; Sarbazi-Azad, H ; Sharif University of Technology
Abstract

This paper proposes two-dimensional directed graphs (or digraphs for short) as a promising alternative to the popular 2D mesh topology for networks-onchip (NoCs). Mesh is the most popular topology for the NoCs, mainly due to its suitability for on-chip implementation and low cost. However, the fact that a digraph offers a lower diameter than its equivalent linear array of equal cost motivated us to evaluate digraphs as the underlying topology of NoCs. This paper introduces a family of NoC topologies based on three well-known digraphs, namely de Bruijn, shuffleexchange, and Kautz. We study topological properties of the proposed topologies. We show that the proposed digraph-based topologies...

#### Linear index coding via graph homomorphism

, Article Proceedings - 2014 International Conference on Control, Decision and Information Technologies, CoDIT 2014 ; 2014 , pp. 158-163 ; ISBN: 9781479967735 ; Siavoshani, M. J ; Sharif University of Technology
Abstract

In [1], [2] it is shown that the minimum broadcast rate of a linear index code over a finite field Fq is equal to an algebraic invariant of the underlying digraph, called minrankq. In [3], it is proved that for F2 and any positive integer k, minrankq(G) ≤ k if and only if there exists a homomorphism from the complement of the graph G to the complement of a particular undirected graph family called 'graph family {Gk}'. As observed in [2], by combining these two results one can relate the linear index coding problem of undirected graphs to the graph homomorphism problem. In [4], a direct connection between linear index coding problem and graph homomorphism problem is introduced. In contrast to...

#### Asymptotic swarm stability of fractional-order swarm systems in the presence of uniform time-delays

, Article International Journal of Control ; Nov , 2015 , Pages 1-10 ; 00207179 (ISSN) ; Tavazoei, M. S ; Sharif University of Technology
Taylor and Francis Ltd
2015

Abstract

In this paper, asymptotic swarm stability of fractional-order swarm systems in the presence of uniform time-delays is studied. By using a frequency domain approach, the necessary and sufficient condition on the values of time-delay is derived to guarantee the asymptotic swarm stability under a directed/undirected interaction graph. Numerical examples are given to confirm the obtained analytical results

#### Development of systematic framework for an intelligent decision support system in gas transmission network

, Article Industrial and Engineering Chemistry Research ; Volume 54, Issue 43 , 2015 , Pages 10768-10786 ; 08885885 (ISSN) ; Bozorgmehry Boozarjomehry, R ; Sharif University of Technology
American Chemical Society
2015

Abstract

In a gas transmission network (GTN), faults can easily propagate due to the interconnections of streams. The main objective of this paper is to develop a systematic framework for an online decision support system (DSS) in order to make the right decisions to get the GTN out of critical conditions (which cannot be handled by the plant controllers) smoothly. One of the key features of the proposed scheme is its lack of dependence on prior knowledge of the fault signals (e.g., number of faults, and their origin). In this article, the GTN is modeled by a fuzzy directed graph (FDG). The proposed approach utilizes a reasoning algorithm based on the deviations that exist in the process variables...

#### Robust decentralized control of consensus-based formations of leader-follower networks with uncertain directed topologies on bounded velocity trajectories unde ned for followers via backstepping method

, Article Scientia Iranica ; Volume 23, Issue 3 , 2016 , Pages 1064-1072 ; 10263098 (ISSN) ; Soltani, A ; Sharif University of Technology
Sharif University of Technology
2016

Abstract

In this paper, decentralized control of formation of a special category of leaderfollower networks on bounded velocity trajectories is addressed. The network of the agents in this study is supposed to have a directed graph with a spanning tree rooted at the leader agent. Moreover, follower agents do not receive online or have offine velocity of the desired trajectory, such as in tracking problem of trajectories which are not predefined or when the total bandwidth is narrow. Furthermore, the leader does not receive any information from any agent and its control is fully centralized. In the present study, formation problem is considered a consensus problem. The controller is designed for...