@phdthesis{Moenius2021,
  author    = {M{\"o}nius, Katja},
  title     = {Algebraic and Arithmetic Properties of Graph Spectra},
  doi       = {10.25972/OPUS-23085},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-230850},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2021},
  abstract  = {In the present thesis we investigate algebraic and arithmetic properties of graph spectra. In particular, we study the algebraic degree of a graph, that is the dimension of the splitting field of the characteristic polynomial of the associated adjacency matrix over the rationals, and examine the question whether there is a relation between the algebraic degree of a graph and its structural properties. This generalizes the yet open question ``Which graphs have integral spectra?'' stated by Harary and Schwenk in 1974. We provide an overview of graph products since they are useful to study graph spectra and, in particular, to construct families of integral graphs. Moreover, we present a relation between the diameter, the maximum vertex degree and the algebraic degree of a graph, and construct a potential family of graphs of maximum algebraic degree. Furthermore, we determine precisely the algebraic degree of circulant graphs and find new criteria for isospectrality of circulant graphs. Moreover, we solve the inverse Galois problem for circulant graphs showing that every finite abelian extension of the rationals is the splitting field of some circulant graph. Those results generalize a theorem of So who characterized all integral circulant graphs. For our proofs we exploit the theory of Schur rings which was already used in order to solve the isomorphism problem for circulant graphs. Besides that, we study spectra of zero-divisor graphs over finite commutative rings. Given a ring \(R\), the zero-divisor graph over \(R\) is defined as the graph with vertex set being the set of non-zero zero-divisors of \(R\) where two vertices \(x,y\) are adjacent if and only if \(xy=0\). We investigate relations between the eigenvalues of a zero-divisor graph, its structural properties and the algebraic properties of the respective ring.},
  subject      = {Algebraische Zahlentheorie},
  language  = {en}
}
@phdthesis{Loeffler2021,
  author    = {L{\"o}ffler, Andre},
  title     = {Constrained Graph Layouts: Vertices on the Outer Face and on the Integer Grid},
  edition   = {1. Auflage},
  publisher = {W{\"u}rzburg University Press},
  address   = {W{\"u}rzburg},
  isbn      = {978-3-95826-146-4},
  doi       = {10.25972/WUP-978-3-95826-147-1},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-215746},
  school      = {W{\"u}rzburg University Press},
  pages     = {viii, 161},
  year      = {2021},
  abstract  = {Constraining graph layouts - that is, restricting the placement of vertices and the routing of edges to obey certain constraints - is common practice in graph drawing. In this book, we discuss algorithmic results on two different restriction types: placing vertices on the outer face and on the integer grid. For the first type, we look into the outer k-planar and outer k-quasi-planar graphs, as well as giving a linear-time algorithm to recognize full and closed outer k-planar graphs Monadic Second-order Logic. For the second type, we consider the problem of transferring a given planar drawing onto the integer grid while perserving the original drawings topology; we also generalize a variant of Cauchy's rigidity theorem for orthogonal polyhedra of genus 0 to those of arbitrary genus.},
  subject      = {Graphenzeichnen},
  language  = {en}
}
@phdthesis{Gregor2008,
  author    = {Gregor, Thomas},
  title     = {{0,1}-Matrices with Rectangular Rule},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-28389},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2008},
  abstract  = {The incidence matrices of many combinatorial structures satisfy the so called rectangular rule, i.e., the scalar product of any two lines of the matrix is at most 1. We study a class of matrices with rectangular rule, the regular block matrices. Some regular block matrices are submatrices of incidence matrices of finite projective planes. Necessary and sufficient conditions are given for regular block matrices, to be submatrices of projective planes. Moreover, regular block matrices are related to another combinatorial structure, the symmetric configurations. In particular, it turns out, that we may conclude the existence of several symmetric configurations from the existence of a projective plane, using this relationship.},
  subject      = {Projektive Ebene},
  language  = {en}
}
@phdthesis{Kramer2004,
  author    = {Kramer, Helmut},
  title     = {Inzidenzmatrizen endlicher projektiver Ebenen},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-11215},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2004},
  abstract  = {Ziel dieser Arbeit ist eine computerunterst{\"u}tzte Suche nach, bis auf Isomorphie, allen projektiven Ebenen zu einer gegebenen Ordnung durch Berechnung ihrer Inzidenzmatrix. Dies gelingt durch geeignete Vorstrukturierung der Matrix mit Hilfe der Doppelordnung bis Ordnung 9 auf einem aktuellen PC. In diesem Zusammenhang ist insbesondere durch einen gen{\"u}gend schnellen Algorithmus das Problem zu l{\"o}sen, ob zwei Inzidenzmatrizen zu derselben projektiven Ebene geh{\"o}ren. Die besondere Struktur, die die berechneten Beispiele von doppelgeordneten Inzidenzmatrizen der desarguesschen Ebenen aufzeigen, wird zudem durch theoretische {\"U}berlegungen untermauert. In einem letzten Kapitel wird noch eine Verbindung der projektiven Ebenen zu besonderen Blockpl{\"a}nen geschaffen.},
  subject      = {Projektive Ebene},
  language  = {de}
}