Tutte decomposition for graphs, weighted graphs, and symmetric matrices

1998 ◽  
pp. 137-146 ◽  
Author(s):  
Sergei K. Lando
Keyword(s):  
Symmetric Matrices ◽  
Weighted Graphs

On the multiplicities of eigenvalues of graphs and their vertex deleted subgraphs: old and new results

2015 ◽  
Vol 30 ◽  
Author(s):  
Slobodan Simic ◽  
Milica Andelic ◽  
Carlos Da Fonseca ◽  
Dejan Zivkovic
Keyword(s):  
Adjacency Matrix ◽  
Simple Graph ◽  
Symmetric Matrices ◽  
Weighted Graphs ◽  
Principal Submatrix ◽  
Eigenvalues Of Graphs

Given a simple graph G, let A(G) be its adjacency matrix. A principal submatrix of A(G) of order one less than the order of G is the adjacency matrix of its vertex deleted subgraph. It is well-known that the multiplicity of any eigenvalue of A(G) and such a principal submatrix can differ by at most one. Therefore, a vertex v of G is a downer vertex (neutral vertex, or Parter vertex) with respect to a fixed eigenvalue μ if the multiplicity of μ in A(G)−v goes down by one (resp., remains the same, or goes up by one). In this paper, we consider the problems of characterizing these three types of vertices under various constraints imposed on graphs being considered, on vertices being chosen and on eigenvalues being observed. By assigning weights to edges of graphs, we generalizeour results to weighted graphs, or equivalently to symmetric matrices.

REMARKS ON THE KIELANOWSKI PARAMETRIZATION OF THE KOBAYASHI-MASKAWA MATRIX

1996 ◽  
Vol 11 (31) ◽  
pp. 2531-2537 ◽  
Author(s):  
TATSUO KOBAYASHI ◽  
ZHI-ZHONG XING
Keyword(s):  
Symmetric Matrices

We study the Kielanowski parametrization of the Kobayashi-Maskawa (KM) matrix V. A new two-angle parametrization is investigated explicitly and compared with the Kielanowski ansatz. Both of them are symmetric matrices and lead to |V13/V23|=0.129. Necessary corrections to the off-diagonal symmetry of V are also discussed.

Converting immanants on skew-symmetric matrices

2021 ◽  
Vol 618 ◽  
pp. 76-96
Author(s):  
M.A. Duffner ◽  
A.E. Guterman ◽  
I.A. Spiridonov
Keyword(s):  
Symmetric Matrices

scholarly journals The location of classified edges due to the change in the geometric multiplicity of an eigenvalue in a tree

2019 ◽  
Vol 7 (1) ◽  
pp. 257-262
Author(s):  
Kenji Toyonaga
Keyword(s):  
Symmetric Matrix ◽  
Symmetric Matrices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Geometric Multiplicity ◽  
Necessary And Sufficient ◽  
Multiplicity Of An Eigenvalue

Abstract Given a combinatorially symmetric matrix A whose graph is a tree T and its eigenvalues, edges in T can be classified in four categories, based upon the change in geometric multiplicity of a particular eigenvalue, when the edge is removed. We investigate a necessary and sufficient condition for each classification of edges. We have similar results as the case for real symmetric matrices whose graph is a tree. We show that a g-2-Parter edge, a g-Parter edge and a g-downer edge are located separately from each other in a tree, and there is a g-neutral edge between them. Furthermore, we show that the distance between a g-downer edge and a g-2-Parter edge or a g-Parter edge is at least 2 in a tree. Lastly we give a combinatorially symmetric matrix whose graph contains all types of edges.

Linear Immanant Converters on Skew-Symmetric Matrices of Order 4

2021 ◽  
Author(s):  
A. E. Guterman ◽  
M. A. Duffner ◽  
I. A. Spiridonov
Keyword(s):  
Symmetric Matrices

scholarly journals Further generalization of symmetric multiplicity theory to the geometric case over a field

2021 ◽  
Vol 9 (1) ◽  
pp. 31-35
Author(s):  
Isaac Cinzori ◽  
Charles R. Johnson ◽  
Hannah Lang ◽  
Carlos M. Saiago
Keyword(s):  
Symmetric Matrices

Abstract Using the recent geometric Parter-Wiener, etc. theorem and related results, it is shown that much of the multiplicity theory developed for real symmetric matrices associated with paths and generalized stars remains valid for combinatorially symmetric matrices over a field. A characterization of generalized stars in the case of combinatorially symmetric matrices is given.

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