Inverse eigenvalue problems and lists of multiplicities of eigenvalues for matrices whose graph is a tree: the case of generalized stars and double generalized stars

Johnson, Charles R.; Duarte, António Leal; Saiago, Carlos M.

doi:10.1016/s0024-3795(03)00582-2

Cited by 52 publications

(33 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This result was further generalized in [8]. More results regarding interlacing equalities in trees can be found in [1,3,4]. In view of the mentioned results, it became natural to consider the case in which G is a tree, as the mentioned results suggest that trees exhibit very special properties related to interlacing equalities.…”

Section: Introductionmentioning

confidence: 85%

The Number of Interlacing Equalities Resulting from Removal of a Vertex from a Tree

Farber¹,

Johnson²,

Zhang³

2015

SIAM J. Discrete Math.

View full text Add to dashboard Cite

Abstract. We consider the set of Hermitian matrices corresponding to a given graph, that is, Hermitian matrices whose nonzero entries correspond to the edges of the graph. When a particular vertex is removed from a graph a number of eigenvalues of the resulting principal submatrix may coincide with eigenvalues of the original Hermitian matrix. Here, we count the maximum number of "interlacing equalities" when the graph is a tree and the original matrix has distinct eigenvalues. We provide an upper bound and lower bound for the count and discuss some conditions under which the count is equal to the upper bound.

show abstract

Section: Introductionmentioning

confidence: 85%

The Number of Interlacing Equalities Resulting from Removal of a Vertex from a Tree

Farber¹,

Johnson²,

Zhang³

2015

SIAM J. Discrete Math.

View full text Add to dashboard Cite

show abstract

“…For other kind of trees, we give bounds on M 2 . In [7], the authors directly computed M 2 for generalized stars (for the notion of generalized star see [6]). …”

Section: Upper and Lower Bounds For Mmentioning

confidence: 99%

The maximum multiplicity and the two largest multiplicities of eigenvalues in a Hermitian matrix whose graph is a tree

Fernandes

2015

Special Matrices

View full text Add to dashboard Cite

Abstract:The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree, M 1 , was understood fully (from a combinatorial perspective) by C.R. Johnson, A. Leal-Duarte (Linear Algebra and Multilinear Algebra 46 (1999) 139-144). Among the possible multiplicity lists for the eigenvalues of Hermitian matrices whose graph is a tree, we focus upon M 2 , the maximum value of the sum of the two largest multiplicities when the largest multiplicity is M 1 . Upper and lower bounds are given for M 2 . Using a combinatorial algorithm, cases of equality are computed for M 2 .

show abstract

“…In [8], the GIEP was solved for S(T ) when T is a generalized star. Moreover, the authors of [8] proved that the IEP for S(G) when G is a generalized star, T , is equivalent to the determination of all possible ordered multiplicity lists of T ; that is, if A ∈ S(T ) has eigenvalues λ 1 < .…”

Section: There Exists a Matrix A In S(t ) With Characteristic Polynomialmentioning

confidence: 99%

“…All the paths, stars and generalized stars (defined in [8]) are also superstars. Sometimes, when T is a superstar …”

Section: There Exists a Matrix A In S(t ) With Characteristic Polynomialmentioning

confidence: 99%

See 1 more Smart Citation

On the inverse eigenvalue problems: the case of superstars

Fernandes¹

2009

ELA

View full text Add to dashboard Cite

Abstract. Let T be a tree and let x 0 be a vertex of T . T is called a superstar with central vertex x 0 if T − x 0 is a union of paths. The General Inverse Eigenvalue Problem for certain trees is partially answered. Using this description, some superstars are presented for which the problem of ordered multiplicity lists and the Inverse Eigenvalue Problem are not equivalent.Key words. Eigenvalues, Tree, Graph, Symmetric matrices. AMS subject classifications. 15A18, 05C38, 05C50. Introduction.Let A = [a ij ] be an n × n real symmetric matrix. We denote by G(A) = (X, U ) the simple graph on n vertices, {1, . . . , n}, such that {i, j} ∈ U , i = j, if and only if a ij = 0. Let A(i) denote the principal submatrix of A obtained by deleting row and column i.Let G = (X, U ) a simple graph, where X = {x 1 , . . . , x n } is the vertex set of G and let S(G) be the set of all n × n real symmetric matrices A such that G(A) ∼ = G. One of the most important problems of Spectral Graph Theory is the General Inverse Eigenvalue Problem for S(G) (GIEP for S(G)):"What are all the real numbers λ 1 ≤ . . . ≤ λ n and µ 1 ≤ . . . ≤ µ n−1 that may occur as the eigenvalues of A and A(i), respectively, as A runs over S(G)?" Another important problem is the Inverse Eigenvalue Problem for S(G) (IEP for S(G));"What are all the real numbers λ 1 ≤ . . . ≤ λ n that may occur as the eigenvalues of A, as A runs over S(G)?" First, we remind the reader of some results concerning the GIEP.Perhaps the most well known result on this subject is the Interlacing Theorem If λ is a real number and A is an n × n real symmetric matrix, we denote by m A (λ) the multiplicity of λ as an eigenvalue of A. As a Corollary of Theorem 1.1 we have the following result: Proposition 1.2.[11] Let A be an n × n Hermitian matrix and let λ be an eigenvalue of A. ThenWhen the graph G is a path, we have the well-known fact: The solution of the GIEP for S(G) when G is a cycle is also well known, see [3,4,5,6].Leal Duarte generalized the converse of Proposition 1.3 to any tree, [13]. Proposition 1.4. Let T be a tree on n vertices and let i be a vertex of T . Let λ 1 < . . . < λ n and µ 1 < . . . < µ n−1 be real numbers. Ifthen there exists a matrix A in S(T ), with eigenvalues λ 1 < . . . < λ n , and such that, A(i) has eigenvalues µ 1 < . . . < µ n−1 .In [7], Johnson and Leal Duarte studied this problem for vertices, of a generic path T , of degree two and solved it for the particular case that occurs when A is a matrix in S(T ) and A(i) has eigenvalues of multiplicity two. "Let G be a simple connected graph G on n vertices, x 0 be a vertex of G of degree k and G 1 , . . . , G k be the connected components of G − x 0 . Let λ 1 , . . . , λ n be real numbers, g 1 (t), . . . , g k (t) be monic polynomials having only real roots and such that Let g(t) be a monic polynomial of degree s + 1. There exists a matrix A in S(T ) with characteristic polynomial if and only if the roots of g(t) strictly interlace those ofThe statement of the previous theorem is shorter when T is a general...

show abstract

Inverse eigenvalue problems and lists of multiplicities of eigenvalues for matrices whose graph is a tree: the case of generalized stars and double generalized stars

Cited by 52 publications

References 8 publications

The Number of Interlacing Equalities Resulting from Removal of a Vertex from a Tree

The Number of Interlacing Equalities Resulting from Removal of a Vertex from a Tree

The maximum multiplicity and the two largest multiplicities of eigenvalues in a Hermitian matrix whose graph is a tree

On the inverse eigenvalue problems: the case of superstars

Contact Info

Product

Resources

About