О P-теменима неких стабала
About P-vertices of some trees
Author
Erić, AleksandraMentor
Lipkovski, AleksandarCommittee members
Fonseca, Carlos Martins daKalajdžić, Gojko
Čukić, Ljubomir
Metadata
Show full item recordAbstract
Ova doktorska disertacija izučava P-temena i P-skupove
nekih nesingularnih acikličnih matrica i takođe nekih singularnih
acikličnih matrica. Ranije je pokazano da svaka
singularna aciklična matrica reda n ima najviše n ¡ 2 P-
temena. Takođe je pokazano da ovo ne važi za nesingularne
aciklične matrice, i to konstrukcijom takvih matrica čiji
pridruženi graf T ima n ¡ 1 ili n P-temena (ove matrica
i dostižu maksimalnu veliqinu P-skupa među acikličnim
nesingularnim matricama čiji je graf T).
U ovoj tezi je data klasifikacija stabala za koje postoji
nesingularna matrica čije je svako teme P-teme. Posebno,
pokazano je da ova stabla moraju imati paran broj temena.
Oba rezultata daju odgovor na otvoreno pitanje koje su postavili
I.-J. Kim i B.L. Shader. Na kraju, izvršena je i klasifikacija
stabala sa ograničenim P-skupom.
Takođe je pokazano da su dvostruke zvezde DSn sa n temena
primer stabala takvih da svaka nesingularna matrica A čiji
je graf DSn ima najvixe n¡2 P-temena. Ovaj primer obezbe...đuje pozitivan odgovor na još jedno pitanje koje su nedavno
otvorili Kim i Shader.
Nedavno je izvršena klasifikacija stabala za koje svaka
pridružena aciklična matrica ima različite sopstvene
vrednosti kada su dijagonalni elementi različiti. U ovom
radu data je analiza maksimalnog broja različitih dijagonalnih elemenata i njihov položaj koji je neophodan da
sačuva traženu višestrukost...
This thesis concerns P-vertices and P-set of non-singular acyclic matrices
A and also singular acyclic matrices. It was shown that each singular matrix
of order n has at most n ¡ 2 P-vertices. Also, it is shown that this does not
hold for non-singular acyclic matrices by constructing non-singular acyclic
matrices whose graphs are T having n¡1 ( or n) P-vertices. These matrices
also achieve maximum size of P-set over non-singular 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 P-vertex. 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 non-singular trees with the size of a P-set bounded are
addressed.
Also, it is shown that double star DSn with n vertices, is an example
of a tree such that, for each non-singular matrix A whose graph is DSn the
...number of P-vertices 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...