Labeling of graphs, sumset of squares of units modulo n and resonance varieties of matroids

This thesis investigates problems in a number of different areas of graph theory and its applications in other areas of mathematics. Motivated by the 1-2-3-Conjecture, we consider the closed distinguishing number of a graph G, denoted by dis[G]. We provide new upper bounds for dis[G] by using the Combinatorial Nullstellensatz. We prove that it is NP-complete to decide for a given planar subcubic graph G, whether dis[G] = 2. We show that for each integer t there is a bipartite graph G such that dis[G] \u3e t. Then some polynomial time algorithms and NP-hardness results for the problem of partitioning the edges of a graph into regular and/or locally irregular subgraphs are presented. We then move on to consider Johnson graphs to find resonance varieties of some classes of sparse paving matroids. The last application we consider is in number theory, where we find the number of solutions of the equation x21 + _ _ _ + x2 k = c, where c 2 Zn, and xi are all units in the ring Zn. Our approach is combinatorial using spectral graph theory

Generalizing Topological Graph Neural Networks with Paths

While Graph Neural Networks (GNNs) have made significant strides in diverse areas, they are hindered by a theoretical constraint known as the 1-Weisfeiler-Lehmann test. Even though latest advancements in higher-order GNNs can overcome this boundary, they typically center around certain graph components like cliques or cycles. However, our investigation goes a different route. We put emphasis on paths, which are inherent in every graph. We are able to construct a more general topological perspective and form a bridge to certain established theories about other topological domains. Interestingly, without any assumptions on graph sub-structures, our approach surpasses earlier techniques in this field, achieving state-of-the-art performance on several benchmarks