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1. Koledin Tamara. Some classes of spectrally constrained graphs.

Degree: PhD, Mathematics, 2013, University of Belgrade

Spectral graph theory is a branch of mathematics that emerged more than sixty years ago, and since then has been continuously developing. Its importance is reflected in many interesting and remarkable applications, esspecially in chemistry, physics, computer sciences and other. Other areas of mathematics, like linear algebra and matrix theory have an important role in spectral graph theory. There are many different matrix representations of a given graph. The ones that have been studied the most are the adjacency matrix and the Laplace matrix, but also the Seidel matrix and the so-called signless Laplace matrix. Basically, the spectral graph theory establishes the connection between some structural properties of a graph and the algebraic properties of its matrix, and considers structural properties that can be described using the properties of the eigenvalues of its matrix. Systematized former results from this vast field of algebraic graph theory can be found in the following monographs: [20], [21], [23] i [58]. This thesis contains original results obtained in several subfields of the spectral graph theory. Those results are presented within three chapters. Each chapter is divided into sections, and some sections into subsections. At the beginning of each chapter (in an appropriate sections), we formulate the problem considered within it, and present the existing results related to this problem, that are necessary for further considerations. All other sections contain only original results. Those results can also be found in the following papers: [3], [4], [47], [48], [49], [50], [51] and [52]. In the first chapter we consider the second largest eigenvalue of a regular graph. There are many results concerning graphs whose second largest eigenvalue is upper bounded by some (relatively small) constant. The second largest eigenvalue plays an important role in determining the structure of regular graphs. There is a known characterization of regular graphs with only one positive eigenvalue (see [20]), and regular graphs with the property λ2 ≤ 1 have also been considered (see [64]). Within this thesis we extend the results given in [64], and we also present some general results concerning the relations between some structural and spectral properties of regular triangle-free graphs. Connected regular graphs with small number of distinct eigenvalues have been extensively studied, since they usually have an interesting (combinatorial) structure. Van Dam and Spence considered the problem of determining the structure of connected regular graphs with exactly four distinct eigenvalues, and they achieved important results presented in papers [27] and [32]. All connected regular bipartite graphs with exactly four distinct eigenvalues are characterized as the incidence graphs of balanced incomplete block designs (see monograph [20]). There are also results concerning regular bipartite graphs with exactly five distinct eigenvalues (see [33]). In this thesis, in the second chapter, we consider regular bipartite graphs with three…

Subjects/Keywords: adjacency matrix; signless Laplace matrix; graph spectrum; signless Laplace spectrum; second largest eigenvalue; regular graph; bipartite graph; nested graph; balanced incomplete block design; partially balanced incomplete block design; matrica susedstva grafa; nenegativna Laplasova matrica grafa; spektar grafa; nenegativni Laplasov spektar grafa; druga sopstvena vrednost; regularan graf; bipartitni graf; ugnežđeni graf; uravnotežena nekompletna blok-šema; delimično uravnotežena nekompletna blok-šema

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APA (6th Edition):

Tamara, K. (2013). Some classes of spectrally constrained graphs. (Doctoral Dissertation). University of Belgrade. Retrieved from http://dx.doi.org/10.2298/BG20130708KOLEDIN ; http://eteze.bg.ac.rs/application/showtheses?thesesId=752 ; https://fedorabg.bg.ac.rs/fedora/get/o:7048/bdef:Content/get ; http://vbs.rs/scripts/cobiss?command=SEARCH&base=99999&select=ID=44728079

Chicago Manual of Style (16th Edition):

Tamara, Koledin. “Some classes of spectrally constrained graphs.” 2013. Doctoral Dissertation, University of Belgrade. Accessed August 08, 2020. http://dx.doi.org/10.2298/BG20130708KOLEDIN ; http://eteze.bg.ac.rs/application/showtheses?thesesId=752 ; https://fedorabg.bg.ac.rs/fedora/get/o:7048/bdef:Content/get ; http://vbs.rs/scripts/cobiss?command=SEARCH&base=99999&select=ID=44728079.

MLA Handbook (7th Edition):

Tamara, Koledin. “Some classes of spectrally constrained graphs.” 2013. Web. 08 Aug 2020.

Vancouver:

Tamara K. Some classes of spectrally constrained graphs. [Internet] [Doctoral dissertation]. University of Belgrade; 2013. [cited 2020 Aug 08]. Available from: http://dx.doi.org/10.2298/BG20130708KOLEDIN ; http://eteze.bg.ac.rs/application/showtheses?thesesId=752 ; https://fedorabg.bg.ac.rs/fedora/get/o:7048/bdef:Content/get ; http://vbs.rs/scripts/cobiss?command=SEARCH&base=99999&select=ID=44728079.

Council of Science Editors:

Tamara K. Some classes of spectrally constrained graphs. [Doctoral Dissertation]. University of Belgrade; 2013. Available from: http://dx.doi.org/10.2298/BG20130708KOLEDIN ; http://eteze.bg.ac.rs/application/showtheses?thesesId=752 ; https://fedorabg.bg.ac.rs/fedora/get/o:7048/bdef:Content/get ; http://vbs.rs/scripts/cobiss?command=SEARCH&base=99999&select=ID=44728079

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