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On linear transformations preserving at least one eigenvalue
, Article Proceedings of the American Mathematical Society ; Volume 132, Issue 6 , 2004 , Pages 1621-1625 ; 00029939 (ISSN) ; Aryapoor, M ; Sharif University of Technology
2004
Abstract
Let F be an algebraically closed field and T: Mn(F) → Mn(F) be a linear transformation. In this paper we show that if T preserves at least one eigenvalue of each matrix, then T preserves all eigenvalues of each matrix. Moreover, for any infinite field F (not necessarily algebraically closed) we prove that if T: Mn(F) → M n(F) is a linear transformation and for any A ∈ Mn(F) with at least an eigenvalue in F, A and T(A) have at least one common eigenvalue in F, then T preserves the characteristic polynomial
Generalizing of numerically solving methods of eigenvalue problems to asymmetrical, damping included case
, Article Proceedings of the 2001 ASME Design Engineering Technical Conference and Computers and Information in Engineering Conference, Pittsburgh, PA, 9 September 2001 through 12 September 2001 ; Volume 1 , 2001 , Pages 401-404 ; Sharif University of Technology
2001
Abstract
In this paper, "Subspace" method is generalized to asymmetrical case. In the new algorithm described here, "Lanczos" method is used to find the first subspace and to solve the eigenvalue problem resulted in generalized subspace method. To solve the standard eigenvalue problem developed by "Lanczos" method "Jacoby" method is used. If eigenvalue problem includes damping matrix, that will be imported in new defined mass and stiffness matrices
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 &...
A note on the algebraic connectivity of a graph and its complement
, Article Linear and Multilinear Algebra ; Volume 69, Issue 7 , 2021 , Pages 1248-1254 ; 03081087 (ISSN) ; Akbari, S ; Sharif University of Technology
Taylor and Francis Ltd
2021
Abstract
For a graph G, let (Formula presented.) 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 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 ; Volume 69, Issue 7 , 2021 , Pages 1248-1254 ; 03081087 (ISSN) ; Akbari, S ; Sharif University of Technology
Taylor and Francis Ltd
2021
Abstract
For a graph G, let (Formula presented.) 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 Informa UK Limited, trading as Taylor & Francis Group
Laplacian eigenvalue distribution and graph parameters
, Article Linear Algebra and Its Applications ; Volume 632 , 2022 , Pages 1-14 ; 00243795 (ISSN) ; Akbari, S ; Fakharan, M. H ; Trevisan, V ; Sharif University of Technology
Elsevier Inc
2022
Abstract
Let G be a graph and I be an interval. In this paper, we present bounds for the number mGI of Laplacian eigenvalues in I in terms of structural parameters of G. In particular, we show that mG(n−α(G),n]≤n−α(G) and mG(n−d(G)+3,n]≤n−d(G)−1, where α(G) and d(G) denote the independence number and the diameter of G, respectively. Also, we characterize bipartite graphs that satisfy mG[0,1)=α(G). Further, in the case of triangle-free or quadrangle-free, we prove that mG(n−1,n]≤1. © 2021 Elsevier Inc
On edge-path eigenvalues of graphs
, Article Linear and Multilinear Algebra ; Volume 70, Issue 15 , 2022 , Pages 2998-3008 ; 03081087 (ISSN) ; Azizi, S ; Ghorbani, M ; Li, X ; Sharif University of Technology
Taylor and Francis Ltd
2022
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