Executive Summary : | In 1978, Lovasz introduced topological methods to study graph colorings through the neighborhood complex of a graph. His work gave birth to the new area of mathematics called topological combinatorics. Since then, many other simplicial complexes have been associated with graphs and studied in detail. These graph complexes are used to find combinatorial information about graphs and appear in various areas of mathematics, including statistical physics, commutative algebra, and combinatorial geometry. This project focuses on two renowned graph complexes: the higher independence complex and the matching complex. The higher independence complexes generalize the well-explored independence complexes of graphs and have shown great potential for many applications. The project aims to explore these complexes from a topological and algebraic point of view, studying the Stanley-Riesner ideal associated with these complexes and establishing new relations between the topological properties of these complexes with combinatorial properties of graphs. The matching complex, which has been extensively studied for certain graph classes, has little information about some well-known graph classes, such as Kneser graphs and rectangular grid graphs. Some partial results are available in the literature for the matching complexes of Kneser and grid graphs, and related results have appeared in reputable journals. The results in this direction are expected to receive significant attention from the mathematics community and be published in reputable journals. |