The number of P-vertices for acyclic matrices of maximum nullity

Abstract For a given n -by- n real symmetric matrix A , if the nullity of the principal submatrix of A , obtained from the deletion of a row and a column of the same index, goes up by one, we call such index P-vertex. In this note, we consider the problem of characterizing the trees for which there is an acyclic matrix of maximum nullity with an extremal number of P-vertices. The range of possible values for the number of P-vertices is determined as well. Some applications of aforementioned results and further properties on the so-called optimal P-sets are also discussed.

[1]  Carlos M. da Fonseca,et al.  The maximum number of P-vertices of some nonsingular double star matrices , 2013, Discret. Math..

[2]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[3]  Z. Du,et al.  The real symmetric matrices of odd order with a P-set of maximum size , 2016 .

[5]  Rosário Fernandes,et al.  The number of P-vertices in a matrix with maximum nullity , 2018, Linear Algebra and its Applications.

[6]  M. Anđelić,et al.  Nonsingular acyclic matrices with full number of P-vertices , 2013 .

[7]  C. Fonseca,et al.  On the number of P-vertices of some graphs , 2011 .

[9]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[10]  Charles R. Johnson,et al.  Hermitian Matrices, Eigenvalue Multiplicities, and Eigenvector Components , 2004, SIAM J. Matrix Anal. Appl..

[12]  Charles R. Johnson,et al.  The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree , 1999 .

[13]  B. Shader,et al.  On Fiedler- and Parter-vertices of acyclic matrices , 2008 .

[14]  Carlos M. da Fonseca,et al.  On the continuity of the maximum size of P-sets of acyclic matrices , 2019, Discret. Math..

[15]  Z. Du,et al.  The acyclic matrices with a P-set of maximum size , 2015 .

[16]  Gerry Wiener.  Spectral multiplicity and splitting results for a class of qualitative matrices , 1984 .

[17]  S. Parter.  On the Eigenvalues and Eigenvectors of a Class of Matrices , 1960 .

[18]  Charles R. Johnson,et al.  The structure of matrices with a maximum multiplicity eigenvalue , 2008 .

[20]  Charles R. Johnson,et al.  The Parter-Wiener Theorem: Refinement and Generalization , 2003, SIAM J. Matrix Anal. Appl..