Gaussian graphical models represent conditional independence relationships in multivariate Gaussian distributions through undirected graphs. Previous studies have explored the algebraic structures of the generators of the vanishing ideal for certain graph types with a specific coloring. Building on this foundation, this thesis investigates cycle graphs in detail, considering all possible colorings of this graph type. The primary objective is to evaluate the validity of a conjecture stating that, in cycle graphs, binomial linear forms are elements of the vanishing ideal if and only if there exists a corresponding symmetry in that graph. Through computational studies and theoretical analyses, the sufficient part of the conjecture is confirmed for all n-cycles, demonstrating that whenever a symmetry exists in a cyclic graph, all binomial linear forms induced by that symmetry lie in the vanishing ideal. The necessary condition, however, is proven only for 3- and 5-cycles. By constructing counterexamples, we show that the necessary condition of the conjecture does not hold for all n-cycles, demonstrating that not all binomial linear elements of the vanishing ideal arise from graph symmetries for all n-cycles. Furthermore, we examine non-binomial linear forms within the vanishing ideal and propose a conjecture regarding their structural properties, which remain to be fully understood. All computations for this thesis are performed using Macaulay2, with the code provided in the appendix to support further research.      «

Gaussian graphical models represent conditional independence relationships in multivariate Gaussian distributions through undirected graphs. Previous studies have explored the algebraic structures of the generators of the vanishing ideal for certain graph types with a specific coloring. Building on this foundation, this thesis investigates cycle graphs in detail, considering all possible colorings of this graph type. The primary objective is to evaluate the validity of a conjecture stating that,...     »