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laplacian-spectrum
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Laplacian spectral characterization of two families of trees
, Article Linear and Multilinear Algebra ; Volume 62, Issue 7 , July , 2014 , Pages 965-977 ; ISSN: 03081087 ; Akbari, S ; Shajari, N ; Sharif University of Technology
Abstract
It is well known that all starlike trees, i.e. trees with exactly one vertex of degree at least three, are determined by their Laplacian spectrum. A double starlike tree is a tree with exactly two vertices of degree at least three. In 2009, the following question was posed: Are all the double starlike trees determined by their Laplacian spectra? In this direction, it was proved that one special double starlike tree Hn(p,p) is determined by its Laplacian spectrum, where Hn(p,q) is a tree obtained by joining p pendant vertices to an end vertex of a path of order and then joining pendant vertices to another end of the path. Also, the banana tree Bn,k is a tree obtained by joining a vertex to...
Spectral characterizations of signed cycles
, Article Linear Algebra and Its Applications ; Volume 553 , 2018 , Pages 307-327 ; 00243795 (ISSN) ; Belardo, F ; Dodongeh, E ; Nematollahi, M. A ; Sharif University of Technology
Elsevier Inc
2018
Abstract
A signed graph is a pair like (G,σ), where G is the underlying graph and σ:E(G)→{−1,+1} is a sign function on the edges of G. In this paper we study the spectral determination problem for signed n-cycles (Cn,σ) with respect to the adjacency spectrum and the Laplacian spectrum. In particular, for the Laplacian spectrum, we prove that balanced odd cycles and unbalanced cycles, denoted, respectively, by C2n+1 + and Cn −, are uniquely determined by their Laplacian spectra (i.e., they are DLS). On the other hand, we determine all Laplacian cospectral mates of the balanced even cycles C2n +, so that we show that C2n + is not DLS. The same problem is then considered for the adjacency spectrum,...
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
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...
Laplacian Eigenvalues of Graphs and their Multiplicity
,
Ph.D. Dissertation
Sharif University of Technology
;
Akbari, Saeed
(Supervisor)
Abstract
In this thesis, we study the multiplicity of the Laplacian eigenvalues of trees. It is known that for trees, integer Laplacian eigenvalues larger than 1 are simple. Here we consider the multiplicities of the other (non- integral) Laplacian eigenvalues.We provide an upper bound and determine the trees of order n that have a multiplicity that is close to the upper bound (n-3)/2 , and emphasize the particular role of the algebraic connectivity.In continuation, let G be a graph and I be an interval. We present bounds for the number m_G I of Laplacian eigenvalues in I in terms of structural parameters of G. In particular, we show that m_G (n-α(G),n]≤ n-α(G) and m_G (n-d(G)+3,n]≤ n-d(G)-1, where...