LEAPS-MPS: Applications of Algebraic and Topological Methods in Graph Theory Throughout the Sciences

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This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). Graphs are used to represent complex networks throughout the sciences; for example, biology, chemistry, computer science, and engineering all use graphical networks to model real-life phenomena. The use of graphs allows for a concise way to model the relationships among a large number of entities in a network, and these relationships can be understood through the structural properties of graphs. Such structural properties of graphs are often quantitative, and the research goal of this project is to study numerous quantitative graph measures. The research results will be applied to complex networks, which arise outside of the mathematical sciences, in order to improve network models. The educational goal of this project is to increase the persistence in the undergraduate mathematics major by creating opportunities that increase both the academic integration and social integration of mathematics majors. Mathematics majors from underrepresented groups will be encouraged to participate with the goal of increasing the number of such individuals who could serve as role models for the scientific workforce of the future. A major goal of this project is to study distance-based graph invariants using various algebraic and topological methods in order to improve models which the graph describes. Some current viewpoints do not account for the symmetries of a graph, which are known to affect certain applications; the first area of focus in this project is to further develop graph invariants to account for such symmetries of graphs. The PI will create generating functions that represent q-analogs of these invariants. The second area of focus in this project is to use embedding techniques to obtain new properties and bounds on graph invariants.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.