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Main Eigenvalues of Graphs and Signed Graphs
, M.Sc. Thesis Sharif University of Technology ; Akbari, Saeed (Supervisor) ; Ghorbani, Ebrahim (Co-Supervisor)
Abstract
Let G be a simple graph. An eigenvalue of G, is a main eigenvalue if the corresponding eigenspace has a non-orthogonal eigenvector to the all-one vector j. A signed graph is a graph with a sign to each edge. If in the adjacency matrix of background graph change elements that corresponded by -1, set -1 and in the other elements don’t make any change, then we reach the sign matrix of a signed graph. By an eigenvalue of a signed graph, we mean an eigenvalue of its sign matrix. In this research, we study main eigenvalues of graphs and signed graphs. At first, we present the necessary and sufficient conditions for any graph which has exactly m main eigenvalues. Then, by introducing and creating...
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 Laplacian Eigenvalues of Signed Graphs
, M.Sc. Thesis Sharif University of Technology ; Akbari, Saieed (Supervisor)
Abstract
A signed graph is a graph with a sign attached to each edge. This article extends some fundamental concepts of the Laplacian matrices from graphs to signed graphs.In particular, the largest Laplacian eigenvalue of a signed graph is investigated,which generalizes the corresponding results on the largest Laplacian eigenvalue of a graph.It is proved that (C2n+1; +) is uniquely determined by its Laplacian spectrum (or is DLS), where (C2n+1; +) is a signed cycle in which all edges have positive sign. On the other hand, we determine all Laplacian cospectral mates of (C2n; +) and hence (C2n; +) is not DLS. Also, we show that for every positive integer n, (Cn;) is DLS. Then, we study the spectrum of...
Strong structural controllability of signed networks
, Article 58th IEEE Conference on Decision and Control, CDC 2019, 11 December 2019 through 13 December 2019 ; Volume 2019-December , 2019 , Pages 4557-4562 ; 07431546 (ISSN); 9781728113982 (ISBN) ; Haeri, M ; Mesbahi, M ; Sharif University of Technology
Institute of Electrical and Electronics Engineers Inc
2019
Abstract
In this paper, we discuss the controllability of a family of linear time-invariant (LTI) networks defined on a signed graph. In this direction, we introduce the notion of positive and negative signed zero forcing sets for the controllability analysis of positive and negative eigenvalues of system matrices with the same sign pattern. A sufficient combinatorial condition that ensures the strong structural controllability of signed networks is then proposed. Moreover, an upper bound on the maximum multiplicity of positive and negative eigenvalues associated with a signed graph is provided. © 2019 IEEE
Signed graphs cospectral with the path
, Article Linear Algebra and Its Applications ; Volume 553 , 2018 , Pages 104-116 ; 00243795 (ISSN) ; Haemers, W. H ; Maimani, H. R ; Parsaei Majd, L ; Sharif University of Technology
Elsevier Inc
2018
Abstract
A signed graph Γ is said to be determined by its spectrum if every signed graph with the same spectrum as Γ is switching isomorphic with Γ. Here it is proved that the path Pn, interpreted as a signed graph, is determined by its spectrum if and only if n≡0,1, or 2 (mod 4), unless n∈{8,13,14,17,29}, or n=3. © 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
On the eigenvalues of signed complete graphs
, Article Linear and Multilinear Algebra ; Volume 67, Issue 3 , 2019 , Pages 433-441 ; 03081087 (ISSN) ; Dalvandi, S ; Heydari, F ; Maghasedi, M ; Sharif University of Technology
Taylor and Francis Ltd
2019
Abstract
Let (Formula presented.) be a signed graph, where G is the underlying simple graph and (Formula presented.) is the sign function on the edges of G. The adjacency matrix of a signed graph has (Formula presented.) or (Formula presented.) for adjacent vertices, depending on the sign of the connecting edges. In this paper, the eigenvalues of signed complete graphs are investigated. We prove that (Formula presented.) and 1 are the eigenvalues of the signed complete graph with the multiplicity at least t if there are (Formula presented.) vertices whose all incident edges are positive or negative, respectively. We study the spectrum of a signed complete graph whose negative edges induce an...
Some criteria for a signed graph to have full rank
, Article Discrete Mathematics ; Volume 343, Issue 8 , 2020 ; Ghafari, A ; Kazemian, K ; Nahvi, M ; Sharif University of Technology
Elsevier B.V
2020
Abstract
A weighted graph Gω consists of a simple graph G with a weight ω, which is a mapping, ω: E(G)→Z∖{0}. A signed graph is a graph whose edges are labelled with −1 or 1. In this paper, we characterize graphs which have a sign such that their signed adjacency matrix has full rank, and graphs which have a weight such that their weighted adjacency matrix does not have full rank. We show that for any arbitrary simple graph G, there is a sign σ so that Gσ has full rank if and only if G has a {1,2}-factor. We also show that for a graph G, there is a weight ω so that Gω does not have full rank if and only if G has at least two {1,2}-factors. © 2020 Elsevier B.V
Signed Complete Graphs with Maximum Index
, Article Discussiones Mathematicae - Graph Theory ; Volume 40, Issue 2 , 2020 , Pages 393-403 ; Dalvandi, S ; Heydari, F ; Maghasedi, M ; Sharif University of Technology
Sciendo
2020
Abstract
Let Γ = (G, σ) be a signed graph, where G is the underlying simple graph and σ E(G) → {-, +} is the sign function on the edges of G. The adjacency matrix of a signed graph has-1 or +1 for adjacent vertices, depending on the sign of the edges. It was conjectured that if is a signed complete graph of order n with k negative edges, k-lt-n-1 and has maximum index, then negative edges form K1 ,k. In this paper, we prove this conjecture if we confine ourselves to all signed complete graphs of order n whose negative edges form a tree of order k + 1. A [1, 2]-subgraph of G is a graph whose components are paths and cycles. Let Γ be a signed complete graph whose negative edges form a [1, 2]-subgraph....
The main eigenvalues of signed graphs
, Article Linear Algebra and Its Applications ; 2020 ; França, F. A. M ; Ghasemian, E ; Javarsineh, M ; de Lima, L. S ; Sharif University of Technology
Elsevier Inc
2020
Abstract
A signed graph Gσ is an ordered pair (V(G),E(G)), where V(G) and E(G) are the set of vertices and edges of G, respectively, along with a map σ that signs every edge of G with +1 or −1. An eigenvalue of the associated adjacency matrix of Gσ, denoted by A(Gσ), is a main eigenvalue if the corresponding eigenspace has a non-orthogonal eigenvector to the all-one vector j. We conjectured that for every graph G≠K2,K4{e}, there is a switching σ such that all eigenvalues of Gσ are main. We show that this conjecture holds for every Cayley graphs, distance-regular graphs, vertex and edge-transitive graphs as well as double stars and paths. © 2020 Elsevier Inc
The main eigenvalues of signed graphs
, Article Linear Algebra and Its Applications ; Volume 614 , 2021 , Pages 270-280 ; 00243795 (ISSN) ; França, F. A. M ; Ghasemian, E ; Javarsineh, M ; de Lima, L. S ; Sharif University of Technology
Elsevier Inc
2021
Abstract
A signed graph Gσ is an ordered pair (V(G),E(G)), where V(G) and E(G) are the set of vertices and edges of G, respectively, along with a map σ that signs every edge of G with +1 or −1. An eigenvalue of the associated adjacency matrix of Gσ, denoted by A(Gσ), is a main eigenvalue if the corresponding eigenspace has a non-orthogonal eigenvector to the all-one vector j. We conjectured that for every graph G≠K2,K4{e}, there is a switching σ such that all eigenvalues of Gσ are main. We show that this conjecture holds for every Cayley graphs, distance-regular graphs, vertex and edge-transitive graphs as well as double stars and paths. © 2020 Elsevier Inc
The main eigenvalues of signed graphs
, Article Linear Algebra and Its Applications ; Volume 614 , 2021 , Pages 270-280 ; 00243795 (ISSN) ; França, F. A. M ; Ghasemian, E ; Javarsineh, M ; de Lima, L. S ; Sharif University of Technology
Elsevier Inc
2021
Abstract
A signed graph Gσ is an ordered pair (V(G),E(G)), where V(G) and E(G) are the set of vertices and edges of G, respectively, along with a map σ that signs every edge of G with +1 or −1. An eigenvalue of the associated adjacency matrix of Gσ, denoted by A(Gσ), is a main eigenvalue if the corresponding eigenspace has a non-orthogonal eigenvector to the all-one vector j. We conjectured that for every graph G≠K2,K4{e}, there is a switching σ such that all eigenvalues of Gσ are main. We show that this conjecture holds for every Cayley graphs, distance-regular graphs, vertex and edge-transitive graphs as well as double stars and paths. © 2020 Elsevier Inc