Genelleştirilmiş Frank matrislerin özellikleri
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Tarih
2021
Yazarlar
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Yayıncı
Aksaray Üniversitesi Fen Bilimleri Enstitüsü
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
Bu çalışmada bilinen Frank matristen yola çıkarak genelleştirilmiş Frank matrisi tanımladık ve bu matrisin determinantını, tersini, karakteristik polinomunu hesapladık. Sturm Teoremini kullanarak belirli koşullar altında genelleştirilmiş Frank matrislerin tüm özdeğerlerinin farklı ve pozitif olduğunu elde ettik. Genelleştirilmiş Frank matrislerin ve Sturm Teoremi ile bulduğumuz sonuçların özel hallerini inceledik. Komşuluk matrisi Frank matris olan Frank graf kavramını ve buna ilaveten Frank enerji kavramını tanımladık. Frank grafın spektral yarıçapı ve ayrıca Frank enerji için bazı sınır değerleri elde ettik. Bu çalışma dört bölümden oluşmaktadır. Birinci bölümde çalışmamızın amacı belirlenmiş ve literatür özetine yer verilmiştir. İkinci bölümde çalışmamızla ilgili bazı temel kavramlar, Frank matris ve özellikleri ile Sturm dizisi hakkında temel bilgiler yer almaktadır. Üçüncü bölümde genelleştirilmiş Frank matrisler ve cebirsel özellikleri, Sturm Teoreminin genelleştirilmiş Frank matrislere uygulanması, genelleştirilmiş Frank matrisin özel hallerini içeren bazı uygulamalara yer verilmiştir. Ayrıca Frank matrisin graf teori ile ilgili uygulamaları, Frank grafın spektral yarıçapı ve enerjisi için bulunan bazı sınırlar yine üçüncü bölümde yer almaktadır. Dördüncü bölüm ise sonuçlar ve önerilerden oluşmaktadır.
In this study, we have defined generalized Frank matrix based on the Frank matrix and have computed the determinant, inverse, characteristic polynomial of this matrix. Using the Sturm Theorem, we obtained that all eigenvalues of generalized Frank matrices are different and positive under certain conditions. We examined the special cases of generalized Frank matrices and the results we found with the Sturm Theorem. We have defined the Frank graph concept which its adjacency matrix equals to the Frank matrix and additionally concept of the Frank energy. We obtained some bounds for spectral radius of the Frank graph and also Frank energy. This study consists of four sections. In the first section, the purpose of our study has been determined and a summary of the literature is included. In the second section, some basic concepts about our study, Frank matrix and its properties, and basic informations about Sturm series are included. In the third section, generalized Frank matrices and their algebraic properties, the application of the Sturm Theorem to the generalized Frank matrices, some applications involving the special cases of the generalized Frank matrix are given. In addition, the applications of the Frank matrix in graph theory, some bounds found for the spectral radius and energy of the Frank graph are also included in the third section. The fourth section consists of results and suggestions.
In this study, we have defined generalized Frank matrix based on the Frank matrix and have computed the determinant, inverse, characteristic polynomial of this matrix. Using the Sturm Theorem, we obtained that all eigenvalues of generalized Frank matrices are different and positive under certain conditions. We examined the special cases of generalized Frank matrices and the results we found with the Sturm Theorem. We have defined the Frank graph concept which its adjacency matrix equals to the Frank matrix and additionally concept of the Frank energy. We obtained some bounds for spectral radius of the Frank graph and also Frank energy. This study consists of four sections. In the first section, the purpose of our study has been determined and a summary of the literature is included. In the second section, some basic concepts about our study, Frank matrix and its properties, and basic informations about Sturm series are included. In the third section, generalized Frank matrices and their algebraic properties, the application of the Sturm Theorem to the generalized Frank matrices, some applications involving the special cases of the generalized Frank matrix are given. In addition, the applications of the Frank matrix in graph theory, some bounds found for the spectral radius and energy of the Frank graph are also included in the third section. The fourth section consists of results and suggestions.
Açıklama
Anahtar Kelimeler
Frank Matris, Determinant, LU Ayrışımı, Karakteristik Polinom, Sturm Dizisi, Frank Graf, Frank Matrix, Determinant, LU Decomposition, Characteristic Polynomial, Sturm Sequence, Frank Graph
