It is known that each eigenvalue of a real symmetric, irreducible, tridiagonal matrix has multipl... more It is known that each eigenvalue of a real symmetric, irreducible, tridiagonal matrix has multiplicity 1. The graph of such a matrix is a path. In this paper, we extend the result by classifying those trees for which each of the associated acyclic matrices has distinct eigenvalues whenever the diagonal entries are distinct.
A woven matrix, W, is a type of block matrix constructed from an m by n (0, 1)-matrix D with row ... more A woven matrix, W, is a type of block matrix constructed from an m by n (0, 1)-matrix D with row sums r 1 , r 2 , . . . , r m and column sums c 1 , c 2 , . . . , c n , r i by r i matrices R i (i = 1, 2, . . . , m), and c j by c j matrices, C j (j = 1, 2, . . . , n). Several properties of the determinant and the spectrum of woven matrices are known. In particular, the determinant of a woven matrix is ±( m i=1 det R i )( n j =1 det C j ). In this paper it is shown that in general the permanent of W is not determined by the permanents of the R i and C j . However, there are instances when
A useful conduit between combinatorics and computational scientific computing is the association ... more A useful conduit between combinatorics and computational scientific computing is the association between graphs and matrices. Trees are fundamental combinatorial objects, and matrices whose associated graph is a tree often arise in applications and computational scientific computing. In this talk we discuss the spectral properties of such matrices.
We give a simple proof that an n × n orthogonal matrix with n ≥ 2 which cannot be written as a di... more We give a simple proof that an n × n orthogonal matrix with n ≥ 2 which cannot be written as a direct sum has at least 4n − 4 nonzero entries.
The SIG-dimension of sphere-of-influence graphs using the sup-norm is investigated. Constructions... more The SIG-dimension of sphere-of-influence graphs using the sup-norm is investigated. Constructions which give an upper bound on the SIG-dimension of arbitrary graphs are given. In addition, lower bounds, which are often tight, on the SIG-dimension of complete multipartite graphs are derived.
It is known that each eigenvalue of a real symmetric, irreducible, tridiagonal matrix has multipl... more It is known that each eigenvalue of a real symmetric, irreducible, tridiagonal matrix has multiplicity 1. The graph of such a matrix is a path. In this paper, we extend the result by classifying those trees for which each of the associated acyclic matrices has distinct eigenvalues whenever the diagonal entries are distinct.
A woven matrix, W, is a type of block matrix constructed from an m by n (0, 1)-matrix D with row ... more A woven matrix, W, is a type of block matrix constructed from an m by n (0, 1)-matrix D with row sums r 1 , r 2 , . . . , r m and column sums c 1 , c 2 , . . . , c n , r i by r i matrices R i (i = 1, 2, . . . , m), and c j by c j matrices, C j (j = 1, 2, . . . , n). Several properties of the determinant and the spectrum of woven matrices are known. In particular, the determinant of a woven matrix is ±( m i=1 det R i )( n j =1 det C j ). In this paper it is shown that in general the permanent of W is not determined by the permanents of the R i and C j . However, there are instances when
A useful conduit between combinatorics and computational scientific computing is the association ... more A useful conduit between combinatorics and computational scientific computing is the association between graphs and matrices. Trees are fundamental combinatorial objects, and matrices whose associated graph is a tree often arise in applications and computational scientific computing. In this talk we discuss the spectral properties of such matrices.
We give a simple proof that an n × n orthogonal matrix with n ≥ 2 which cannot be written as a di... more We give a simple proof that an n × n orthogonal matrix with n ≥ 2 which cannot be written as a direct sum has at least 4n − 4 nonzero entries.
The SIG-dimension of sphere-of-influence graphs using the sup-norm is investigated. Constructions... more The SIG-dimension of sphere-of-influence graphs using the sup-norm is investigated. Constructions which give an upper bound on the SIG-dimension of arbitrary graphs are given. In addition, lower bounds, which are often tight, on the SIG-dimension of complete multipartite graphs are derived.
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Papers by Bryan Shader