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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
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 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
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