The Parter--Wiener Theorem: Refinement and Generalization
From MaRDI portal
Publication:4443829
DOI10.1137/S0895479801393320zbMath1067.15003OpenAlexW1985451918MaRDI QIDQ4443829
António Leal-Duarte, Charles R. Johnson, Carlos M. Saiago
Publication date: 19 January 2004
Published in: SIAM Journal on Matrix Analysis and Applications (Search for Journal in Brave)
Full work available at URL: https://doi.org/10.1137/s0895479801393320
eigenvaluestreesmultiplicity of eigenvaluesHermitian matrixtridiagonal matricesvertex degreesParter vertices
Graphs and linear algebra (matrices, eigenvalues, etc.) (05C50) Eigenvalues, singular values, and eigenvectors (15A18) Hermitian, skew-Hermitian, and related matrices (15B57)
Related Items
Inverse eigenvalue problems for two special acyclic matrices ⋮ The change in multiplicity of an eigenvalue due to adding or removing edges ⋮ Graphs with eigenvalue \(-1\) of multiplicity \(2 \theta (G)+ \rho (G) -1\) ⋮ The change in eigenvalue multiplicity associated with perturbation of a diagonal entry ⋮ Normal matrices subordinate to a tree and flat portions of the field of values ⋮ The singular acyclic matrices with the second largest number of P-vertices ⋮ Inverse eigenvalue problems for acyclic matrices whose graph is a dense centipede ⋮ Questions, conjectures, and data about multiplicity lists for trees ⋮ Multiplicities: Adding a Vertex to a Graph ⋮ Minimum rank and maximum eigenvalue multiplicity of symmetric tree sign patterns ⋮ Converse to the Parter--Wiener theorem: the case of non-trees ⋮ The trees for which maximum multiplicity implies the simplicity of other eigenvalues ⋮ Leaders in multi-agent controllability under consensus algorithm and tree topology ⋮ Further generalization of symmetric multiplicity theory to the geometric case over a field ⋮ Geometric Parter-Wiener, etc. theory ⋮ Changes in vertex status and the fundamental decomposition of a tree relative to a multiple (parter) eigenvalue ⋮ Unordered multiplicity lists of a class of binary trees ⋮ Acyclic matrices with a small number of distinct eigenvalues ⋮ The change in multiplicity of an eigenvalue of a Hermitian matrix associated with the removal of an edge from its graph ⋮ On the number of P-vertices of some graphs ⋮ The multiplicity of eigenvalues of unicyclic graphs ⋮ Ordered multiplicity lists for eigenvalues of symmetric matrices whose graph is a linear tree ⋮ Multiplicity lists for symmetric matrices whose graphs have few missing edges ⋮ The number of P-vertices for acyclic matrices with given nullity ⋮ A relation between multiplicity of nonzero eigenvalues of trees and their matching numbers ⋮ The characterization of the minimal weighted acyclic graphs ⋮ The implicit construction of multiplicity lists for classes of trees and verification of some conjectures ⋮ Extremal realization spectra by two acyclic matrices whose graphs are caterpillars ⋮ The minimum number of multiplicity 1 eigenvalues among real symmetric matrices whose graph is a 2-linear tree ⋮ Line graphs of trees with the largest eigenvalue multiplicity ⋮ Diminimal families of arbitrary diameter ⋮ A characterization of trees with eigenvalue multiplicity one less than their number of pendant vertices ⋮ Eigenvalue multiplicity of graphs with given cyclomatic number and given number of quasi-pendant vertices ⋮ Null vectors, Schur complements, and Parter vertices ⋮ The minimum number of multiplicity 1 eigenvalues among real symmetric matrices whose graph is a linear tree ⋮ Branch duplication for the construction of multiple eigenvalues in an Hermitian matrix whose graph is a tree ⋮ Gallai-Edmonds structure theorem for weighted matching polynomial ⋮ 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. ⋮ Sets of Parter vertices which are Parter sets ⋮ On Fiedler- and Parter-vertices of acyclic matrices ⋮ On the multiplicities of eigenvalues of a Hermitian matrix whose graph is a tree ⋮ The structure of matrices with a maximum multiplicity eigenvalue ⋮ Change in vertex status after removal of another vertex in the general setting ⋮ The effect of perturbation of an off-diagonal entry pair on the geometric multiplicity of an eigenvalue ⋮ A zero forcing technique for bounding sums of eigenvalue multiplicities ⋮ The inverse eigenvalue problem of a graph: multiplicities and minors ⋮ The number of \(P\)-vertices in a matrix with maximum nullity ⋮ The classification of edges and the change in multiplicity of an eigenvalue of a real symmetric matrix resulting from the change in an edge value ⋮ The change in multiplicity of an eigenvalue of a real symmetric matrix resulting from the changes in edge values around a classified vertex in a tree ⋮ Undirected graphs of Hermitian matrices that admit only two distinct eigenvalues ⋮ On edge star sets in trees ⋮ Eigenvalue assignments and the two largest multiplicities in a Hermitian matrix whose graph is a tree ⋮ Patterns with several multiple eigenvalues ⋮ The maximum multiplicity and the two largest multiplicities of eigenvalues in a Hermitian matrix whose graph is a tree ⋮ Trees and acyclic matrices over arbitrary fields ⋮ Diameter minimal trees ⋮ Classification of vertices and edges with respect to the geometric multiplicity of an eigenvalue in a matrix, with a given graph, over a field ⋮ The minimum rank of symmetric matrices described by a graph: a survey ⋮ The number of distinct eigenvalues for which an index decreases multiplicity ⋮ The minimum number of eigenvalues of multiplicity one in a diagonalizable matrix, over a field, whose graph is a tree ⋮ The acyclic matrices with a P-set of maximum size ⋮ Numerical enclosure for multiple eigenvalues of an Hermitian matrix whose graph is a tree ⋮ The number of P-vertices of singular acyclic matrices: an inverse problem ⋮ Maximal P-sets of matrices whose graph is a tree ⋮ Smith normal form and acyclic matrices ⋮ The location of classified edges due to the change in the geometric multiplicity of an eigenvalue in a tree ⋮ Diagonalizable matrices whose graph is a tree: the minimum number of distinct eigenvalues and the feasibility of eigenvalue assignments ⋮ Non-singular acyclic matrices ⋮ Implicit construction of multiple eigenvalues for trees ⋮ Unnamed Item ⋮ The real symmetric matrices of odd order with a P-set of maximum size ⋮ The maximum multiplicity of the largest \(k\)-th eigenvalue in a matrix whose graph is acyclic or unicyclic ⋮ The inverse eigenvalue problem for linear trees ⋮ The minimum number of multiplicity 1 eigenvalues among real symmetric matrices whose graph is a nonlinear tree ⋮ The multiplicity of eigenvalues of trees ⋮ The number of P-vertices for acyclic matrices of maximum nullity ⋮ The Number of Interlacing Equalities Resulting from Removal of a Vertex from a Tree