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#### The algebraic connectivity of a graph and its complement

, Article Linear Algebra and Its Applications ; Volume 555 , 2018 , Pages 157-162 ; 00243795 (ISSN) ; Akbari, S ; Moghaddamzadeh, M. J ; Mohar, B ; Sharif University of Technology
Elsevier Inc
2018

Abstract

For a graph G, let λ2(G) denote its second smallest Laplacian eigenvalue. It was conjectured that λ2(G)+λ2(G‾)≥1, where G‾ is the complement of G. In this paper, it is shown that max{λ2(G),λ2(G‾)}≥[Formula presented]. © 2018 Elsevier Inc

#### On the largest eigenvalue of signed unicyclic graphs

, Article Linear Algebra and Its Applications ; Volume 581 , 2019 , Pages 145-162 ; 00243795 (ISSN) ; Belardo, F ; Heydari, F ; Maghasedi, M ; Souri, M ; Sharif University of Technology
Elsevier Inc
2019

Abstract

Signed graphs are graphs whose edges get signs ±1 and, as for unsigned graphs, they can be studied by means of graph matrices. Here we focus our attention to the largest eigenvalue, also known as the index of the adjacency matrix of signed graphs. Firstly we give some general results on the index variation when the corresponding signed graph is perturbed. Also, we determine signed graphs achieving the minimal or the maximal index in the class of unbalanced unicyclic graphs of order n≥3. © 2019

#### Some results on the Laplacian spread conjecture

, Article Linear Algebra and Its Applications ; Volume 574 , 2019 , Pages 22-29 ; 00243795 (ISSN) ; Akbari, S ; Sharif University of Technology
Elsevier Inc
2019

Abstract

For a graph G of order n, let λ 2 (G) denote its second smallest Laplacian eigenvalue. It was conjectured that λ 2 (G)+λ 2 (G‾)≥1, where G‾ is the complement of G. For any x∈R n , let ∇ x ∈R (n2) be the vector whose {i,j}-th entry is |x i −x j |. In this paper, we show the aforementioned conjecture is equivalent to prove that every two orthonormal vectors f,g∈R n with zero mean satisfy ‖∇ f −∇ g ‖ 2 ≥2. In this article, it is shown that for the validity of the conjecture it suffices to prove that the conjecture holds for all permutation graphs. © 2019 Elsevier Inc

#### On edge star sets in trees

, Article Discrete Mathematics ; Volume 311, Issue 13 , July , 2011 , Pages 1172-1178 ; 0012365X (ISSN) ; Ghorbani, E ; Mahmoodi, A ; Sharif University of Technology
2011

Abstract

Let A be a Hermitian matrix whose graph is G (i.e. there is an edge between the vertices i and j in G if and only if the (i,j) entry of A is non-zero). Let λ be an eigenvalue of A with multiplicity mA(λ). An edge e=ij is said to be Parter (resp., neutral, downer) for λ,A if mA(λ)-mA-e(λ) is negative (resp., 0, positive ), where A-e is the matrix resulting from making the (i,j) and (j,i) entries of A zero. For a tree T with adjacency matrix A a subset S of the edge set of G is called an edge star set for an eigenvalue λ of A, if |S|=mA(λ) and A-S has no eigenvalue λ. In this paper the existence of downer edges and edge star sets for non-zero eigenvalues of the adjacency matrix of a tree is...

#### A note on the algebraic connectivity of a graph and its complement

, Article Linear and Multilinear Algebra ; 2019 ; 03081087 (ISSN) ; Akbari, S ; Sharif University of Technology
Taylor and Francis Ltd
2019

Abstract

For a graph G, let λ2(G) denote its second smallest Laplacian eigenvalue. It was conjectured that (Formula presented.), where (Formula presented.) is the complement of G. In this paper, it is shown that (Formula presented.). © 2019, © 2019 Informa UK Limited, trading as Taylor & Francis Group

#### A note on the algebraic connectivity of a graph and its complement

, Article Linear and Multilinear Algebra ; 2019 ; 03081087 (ISSN) ; Akbari, S
Taylor and Francis Ltd
2019

Abstract

For a graph G, let λ2(G) denote its second smallest Laplacian eigenvalue. It was conjectured that (Formula presented.), where (Formula presented.) is the complement of G. In this paper, it is shown that (Formula presented.). © 2019, © 2019 Informa UK Limited, trading as Taylor & Francis Group

#### Trees with a large Laplacian eigenvalue multiplicity

, Article Linear Algebra and Its Applications ; Volume 586 , 2020 , Pages 262-273 ; van Dam, E. R ; Fakharan, M. H ; Sharif University of Technology
Elsevier Inc
2020

Abstract

In this paper, we study the multiplicity of the Laplacian eigenvalues of trees. It is known that for trees, integer Laplacian eigenvalues larger than 1 are simple and also the multiplicity of Laplacian eigenvalue 1 has been well studied before. Here we consider the multiplicities of the other (non-integral) Laplacian eigenvalues. We give an upper bound and determine the trees of order n that have a multiplicity that is close to the upper bound [Formula presented], and emphasize the particular role of the algebraic connectivity. © 2019 Elsevier Inc

#### On edge-path eigenvalues of graphs

, Article Linear and Multilinear Algebra ; 2020 ; Azizi, S ; Ghorbani, M ; Li, X ; Sharif University of Technology
Taylor and Francis Ltd
2020

Abstract

Let G be a graph with vertex set (Formula presented.) and (Formula presented.) be an (Formula presented.) matrix whose (Formula presented.) -entry is the maximum number of internally edge-disjoint paths between (Formula presented.) and (Formula presented.), if (Formula presented.), and zero otherwise. Also, define (Formula presented.), where D is a diagonal matrix whose i-th diagonal element is the number of edge-disjoint cycles containing (Formula presented.), whose (Formula presented.) is a multiple of J−I. Among other results, we determine the spectrum and the energy of the matrix (Formula presented.) for an arbitrary bicyclic graph G. © 2020 Informa UK Limited, trading as Taylor &...

#### On graphs whose star sets are (co-)cliques

, Article Linear Algebra and Its Applications ; Volume 430, Issue 1 , 2009 , Pages 504-510 ; 00243795 (ISSN) ; Ghorbani, E ; Mahmoodi, A ; Sharif University of Technology
Abstract

In this paper we study graphs all of whose star sets induce cliques or co-cliques. We show that the star sets of every tree for each eigenvalue are independent sets. Among other results it is shown that each star set of a connected graph G with three distinct eigenvalues induces a clique if and only if G = K1, 2 or K2, ..., 2. It is also proved that stars are the only graphs with three distinct eigenvalues having a star partition with independent star sets. © 2008 Elsevier Inc. All rights reserved

#### Optimal ground state energy of two-phase conductors

, Article Electronic Journal of Differential Equations ; Vol. 2014 , 2014 ; ISSN: 10726691 ; Yousefnezhad, M ; Sharif University of Technology
Abstract

We consider the problem of distributing two conducting materials in a ball with xed proportion in order to minimize the rst eigenvalue of a Dirichlet operator. It was conjectured that the optimal distribution consists of putting the material with the highest conductivity in a ball around the center. In this paper, we show that the conjecture is false for all dimensions greater than or equal to two

#### Asymptotic eigenvectors, topological patterns and recurrent networks

, Article Proceedings of the Romanian Academy Series A - Mathematics Physics Technical Sciences Information Science ; Volume 14, Issue 2 , 2013 , Pages 95-100 ; 14549069 (ISSN) ; Sharif University of Technology
2013

Abstract

The notions of asymptotic eigenvectors and asymptotic eigenvalues are defined. Based on these notions a special probability rule for pattern selection in a Hopfield type dynamics is introduced. The underlying network is considered to be a d-regular graph, where d is an integer denoting the number of nodes connected to each neuron. It is shown that as far as the degree d is less than a critical value dc, the number of stored patterns with m μ = O(1) can be much larger than that in a standard recurrent network with Bernouill random patterns. As observed in [4] the probability rule we study here turns out to be related to the spontaneous activity of the network. So our result might be an...

#### Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem

, Article Journal of Functional Analysis ; Volume 261, Issue 12 , 2011 , Pages 3419-3436 ; 00221236 (ISSN) ; Sharif University of Technology
2011

Abstract

In this paper, we find upper bounds for the eigenvalues of the Laplacian in the conformal class of a compact Riemannian manifold (M,g). These upper bounds depend only on the dimension and a conformal invariant that we call "min-conformal volume". Asymptotically, these bounds are consistent with the Weyl law and improve previous results by Korevaar and Yang and Yau. The proof relies on the construction of a suitable family of disjoint domains providing supports for a family of test functions. This method is interesting for itself and powerful. As a further application of the method we obtain an upper bound for the eigenvalues of the Steklov problem in a domain with C1 boundary in a complete...

#### A quasi-newtonian approach to bohmian mechanics II: inherent quantization

, Article Annales de la Fondation Louis de Broglie ; Volume 34, Issue 2 , 2009 , Pages 165-181 ; 01824295 (ISSN) ; Karamian, M ; Golshani, M ; Sharif University of Technology
2009

Abstract

In a previous paper, we obtained the functional form of quantum potential by a quasi-Newtonian approach and without appealing to the wave function. We also described briefly the characteristics ofthis approach to the Bohmian mechanics. In this article, we consider the quantization problem and we show that the 'eigenvalue postulate' is a natural consequence of continuity condition and there is no need for postulating that the spectrum of energy and angular momentum are eigenvalues of their relevant operators. In other words, the Bohmian mechanics predicts the 'eigenvalue postulate'

#### The multiplicity of Laplacian eigenvalue two in unicyclic graphs

, Article Linear Algebra and Its Applications ; Vol. 445 , 2014 , pp. 18-28 ; Kiani, D ; Mirzakhah, M ; Sharif University of Technology
Abstract

Let G be a graph and L(G) be the Laplacian matrix of G. In this paper, we explicitly determine the multiplicity of Laplacian eigenvalue 2 for any unicyclic graph containing a perfect matching

#### A relation between the Laplacian and signless Laplacian eigenvalues of a graph

, Article Journal of Algebraic Combinatorics ; Volume 32, Issue 3 , 2010 , Pages 459-464 ; 09259899 (ISSN) ; Ghorbani, E ; Koolen, J. H ; Oboudi, M. R ; Sharif University of Technology
2010

Abstract

Let G be a graph of order n such that ∑n i=0(-1) iailambdan-i and ∑n i=0(-1) iailambdan-i are the characteristic polynomials of the signless Laplacian and the Laplacian matrices of G, respectively. We show that a i ≥b i for i=0,1,⋯,n. As a consequence, we prove that for any α, 0<α≤1, if q 1,⋯,q n and μ 1,⋯,μ n are the signless Laplacian and the Laplacian eigenvalues of G, respectively, then q 1 alpha+⋯+qα n≥μ α 1+⋯+μα n

#### A methodology for analyzing the transient reliability of systems with identical components and identical repairmen

, Article Scientia Iranica ; Volume 14, Issue 1 , 2007 , Pages 72-77 ; 10263098 (ISSN) ; Ghassemi Tari, F ; Sharif University of Technology
Sharif University of Technology
2007

Abstract

In this paper, the Markov models, eigenvectors and eigenvalue concepts are used to propose a methodology for analyzing the transient reliability of a system with identical components and identical repairmen. The components of the systems under consideration can have two distinct configurations, namely; they can be arranged in series or in parallel. A third case is also considered, in which the system is up (good) if k-out-of-n components are good. For all three cases, a procedure is proposed for calculating the transient probability of the system availability and the duration of the system to reach the steady state. © Sharif University of Technology, February 2007

#### Out-of-plane buckling of Y-braced frames with rigid joints

, Article Proceedings of the Institution of Civil Engineers: Structures and Buildings ; Volume 166, Issue 1 , 2013 , Pages 28-37 ; 09650911 (ISSN) ; Vafai, A ; Kazemi, M.T ; Sharif University of Technology
2013

Abstract

Because of the complicated buckling characteristics of Y-shaped bracings, calculation of their strength is beyond routine engineering procedures. A method based on slope-deflection equations incorporating stability functions has been used for computation of buckling eigenvalues and effective length factors. Lateral strength is computed based on the least buckling strength of bracing members. This study is limited to bracings with similar sections and fixed end connections resisting out-of-plane rotation. Lateral strengths predicted by the analytical method for two cases are compared with experimental results on full-scale specimens. It is shown that the proposed analytical method predicts...

#### Eigenvalue estimation of the exponentially windowed sample covariance matrices

, Article IEEE Transactions on Information Theory ; Volume 62, Issue 7 , 2016 , Pages 4300-4311 ; 00189448 (ISSN) ; Gazor, S ; Bastani, M. H ; Sharifitabar, M ; Sharif University of Technology
Institute of Electrical and Electronics Engineers Inc
2016

Abstract

In this paper, we consider an exponentially windowed sample covariance matrix (EWSCM) and propose an improved estimator for its eigenvalues. We use new advances in random matrix theory, which describe the limiting spectral distribution of the large dimensional doubly correlated Wishart matrices to find the support and distribution of the eigenvalues of the EWSCM. We then employ the complex integration and residue theorem to design an estimator for the eigenvalues, which satisfies the cluster separability condition, assuming that the eigenvalue multiplicities are known. We show that the proposed estimator is consistent in the asymptotic regime and has good performance in finite sample size...

#### Dynamic behavior and transient stability analysis of fixed speed wind turbines

, Article Renewable Energy ; Volume 34, Issue 12 , 2009 , Pages 2613-2624 ; 09601481 (ISSN) ; Parniani, M ; Sharif University of Technology
Abstract

This paper analytically investigates the dynamic behavior of fixed speed wind turbines (FSWTs) under wind speed fluctuations and system disturbances, and identifies the nature of transient instability and system variables involved in the instability. The nature of transient instability in FSWT is not similar to synchronous generators in which the cause of instability is rotor angle instability. In this paper, the study of dynamic behavior includes modal and sensitivity analysis, dynamic behavior analysis under wind speed fluctuation, eigenvalue tracking, and using it to characterize the instability mode, and investigating possible outcomes of instability. The results of theoretical studies...

#### Effect of unitary transformation on Bayesian information criterion for source numbering in array processing

, Article IET Signal Processing ; Volume 13, Issue 7 , 2019 , Pages 670-678 ; 17519675 (ISSN) ; Aref, M. R ; Razzazi, F ; Sharif University of Technology
Institution of Engineering and Technology
2019

Abstract

An approach based on unitary transformation for the problem of estimating the number of signals is proposed in this study. Among the information theoretic criteria, the authors focus on the conventional Bayesian information criterion (BIC) in the presence of a uniform linear array. The sample covariance matrix of this array is transformed into the real symmetric one by using a unitary transformation. This real symmetric matrix has real eigenvalues and eigenvectors. Therefore its eigenvalue decomposition needs only real computations. Since the eigenvalues of this real symmetric matrix are equal to the eigenvalues of the sample covariance matrix, by replacing them in BIC formula, the term...