Abstract:

This thesis concerns Pvertices and Pset of nonsingular acyclic matrices
A and also singular acyclic matrices. It was shown that each singular matrix
of order n has at most n ¡ 2 Pvertices. Also, it is shown that this does not
hold for nonsingular acyclic matrices by constructing nonsingular acyclic
matrices whose graphs are T having n¡1 ( or n) Pvertices. These matrices
also achieve maximum size of Pset over nonsingular acyclic matrices whose
graphs are T.
In this thesis, there is classi¯cation of the trees for which there is non
singular matrix where each vertex is Pvertex. In particular, it is shown
that such trees have an even number of vertices. Both results provide answer
to questions proposed by I.J. Kim and B. L. Shader. In the end, related
classi¯cations on nonsingular trees with the size of a Pset bounded are
addressed.
Also, it is shown that double star DSn with n vertices, is an example
of a tree such that, for each nonsingular matrix A whose graph is DSn the
number of Pvertices of A is less than n¡2. This example provides a positive
answer to a question proposed recently by Kim and Shader.
A recent classi¯cation of those trees for which each of associated acyclic
matrices has distinct eigenvalues whenever the diagonal entries are distinct
was established. Here is analyze of maximum number of distinct diagonal
entries, and corresponding location, in order to preserve that multiplicity
characterization.
Recently, the multiplicities of eigenvalues of ©binary tree was analyzed.
This paper carry this discussion forward extending their results to larger
family of trees, namely, the wide double path, a tree consisting of two paths
that are joined by another path.
Some introductory considerations for dumbbell graphs are mentioned re
garding the maximum multiplicity of the eigenvalues. 