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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
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
Graphs whose spectrum determined by non-constant coefficients
, Article Electronic Notes in Discrete Mathematics ; Vol. 45 , 2014 , pp. 29-34 ; ISSN: 15710653 ; Kiani, D ; Mirzakhah, M ; Sharif University of Technology
Abstract
Let G be a graph and M be a matrix associated with G whose characteristic polynomial is M(G,x)=∑i=0nαi(G)xn-i. We say that the spectrum of G is determined by non-constant coefficients (simply M-SDNC), if for any graph H with ai(H)=ai(G), 0≤i≤n-1, then Spec(G)=Spec(H) (if M is the adjacency matrix or the Laplacian matrix of G, then G is called an A-SDNC graph or L-SDNC graph). In this paper, we study some properties of graphs which are A-SDNC or L-SDNC. Among other results, we prove that the path of order at least five is L-SDNC and moreover stars of order at least five are both A-SDNC and L-SDNC. Furthermore, we construct infinitely many trees which are not A-SDNC graphs. More precisely, we...
On the Density Matrix of Graphs
, M.Sc. Thesis Sharif University of Technology ; Daneshgar, Amir (Supervisor)
Abstract
Density matrix of graphs as defined -for the first time- in [S. Braunstein and et al. The laplacian of a graph as a density matrix, Annals of Combinatorics, (2006)], is obtained through dividing the Laplacian matrix by the degree sum. This matrix is also semi-positive and trace one. Therefore one may talk about the Von Neumann entropy of this matrix. In [F. Passerini, S. Severini. Quantifying complexity in networks: The Von Neumann entropy. IJATS, (2009)], authors have claimed that this quantity can be consisered as a measure of regularity. Here, using a geometric interpretation of Von Neumann entropy, expresed in [G. Mitchison, R. Jozsa, Towards a geometrical interpretation of quantum...
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
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
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
A lower bound for algebraic connectivity based on the connection-graph- stability method
, Article Linear Algebra and Its Applications ; Volume 435, Issue 1 , Sep , 2011 , Pages 186-192 ; 00243795 (ISSN) ; Jalili, M ; Hasler, M ; Sharif University of Technology
2011
Abstract
This paper introduces the connection-graph-stability method and uses it to establish a new lower bound on the algebraic connectivity of graphs (the second smallest eigenvalue of the Laplacian matrix of the graph) that is sharper than the previously published bounds. The connection-graph-stability score for each edge is defined as the sum of the lengths of the shortest paths making use of that edge. We prove that the algebraic connectivity of the graph is bounded below by the size of the graph divided by the maximum connection-graph-stability score assigned to the edges
Spectral Graph Partitioning
, M.Sc. Thesis Sharif University of Technology ; Daneshgar, Amir (Supervisor)
Abstract
Graph partitioning, or graph clustering, is an essential researa problem in many areas. In this thesis, we focus on the partitioning problem of unweighted undirected graph, that is, graphs for which the weight of all edges is 1. We will investigate spectral methods for solving the graph partitioning and we compare them. In addition to theoretical analysis,We also implement some of spectral algorithms in matlab and apply them on standard graph data sets. Finally, the experimental
results obtained are offering
results obtained are offering