Graphs with Few Eigenvalues. An Interplay between Combinatorics and Algebra.

Abstract: Two standard matrix representations of a graph are the adjacency matrix and the Laplace matrix. The eigenvalues of these matrices are interesting parameters of the graph. Graphs with few eigenvalues in general have nice combinatorial properties and a rich structure. A well investigated family of such graphs comprises the strongly regular graphs (the regular graphs with three eigenvalues), and we may see other graphs with few eigenvalues as algebraic generalizations of such graphs. We study the (nonregular) graphs with three adjacency eigenvalues, graphs with three Laplace eigenvalues, and regular graphs with four eigenvalues. The last ones are also studied in relation with three-class association schemes. We also derive bounds on the diameter and on the size of special subsets in terms of the eigenvalues of the graph. Included are lists of feasible parameter sets of graphs with three Laplace eigenvalues, regular graphs with four eigenvalues, and three-class association schemes.

A Geometric Approach to the Projective Tensor Norm

The main focus of this thesis is on the projective norm on finite-dimensional real or complex tensor products. There are various mathematical subjects with relations to the projective norm. For instance, it appears in the context of operator algebras or in quantum physics. The projective norm on multipartite tensor products is considered to be less accessible. So we use a method from convex algebraic geometry to approximate the projective unit ball by convex supersets, so-called theta bodies. For real multipartite tensor products we obtain theta bodies which are close to the projective unit ball, leading to a generalisation of the Schmidt decomposition. In a second step the method is applied to complex tensor products, in a third step to separable states. In a more general context, the projective norm can be related to binomial ideals, especially to so-called Hibi relations. In this respect, we also focus on a generalisation of the projective unit ball, here called Hibi body, and its theta bodies. It turns out that many statements also hold in this general context

Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group