Tutte decomposition for graphs, weighted graphs, and symmetric matrices

Author(s):  
Sergei K. Lando
Author(s):  
Slobodan Simic ◽  
Milica Andelic ◽  
Carlos Da Fonseca ◽  
Dejan Zivkovic

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.


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

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.


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

2019 ◽  
Vol 7 (1) ◽  
pp. 257-262
Author(s):  
Kenji Toyonaga

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.


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

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

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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