Research Projects

In this project, we will primarily focus on spectral inequalities and matrix inequalities, which occupy a central place in the study of matrices and graphs. These inequalities are essential tools in mathematical analysis and have wide-ranging applications in spectral graph theory, linear algebra, control theory, and network science. Typically, such inequalities provide lower or upper bounds for specific eigenvalues of matrices associated with graphs, such as adjacency, Laplacian, or $A_\alpha$ matrices. These bounds help us understand the structure and behavior of graphs through their spectral properties. A key objective of this project is to go beyond classical results and derive new or sharper inequalities for eigenvalues-either individually or in combinations such as sums or variances. In particular, we aim to explore inequalities involving spectral moments (e.g., sum of squares or higher powers of eigenvalues) and spread-type quantities, which can give deeper insights into the distribution of eigenvalues. In addition to eigenvalue-based inequalities, we will also investigate inequalities that bound certain graph invariants-such as graph energy, spectral radius, or algebraic connectivity-in terms of eigenvalues and possibly other structural parameters (e.g., degrees or number of vertices). These types of inequalities play a crucial role in spectral graph theory as they connect the algebraic properties of a graph with its combinatorial or topological structure. Another important component of this work will be the study of extremal graphs-graphs that maximize or minimize a given spectral invariant within a defined class (e.g., trees, bipartite graphs, or regular graphs). Identifying such graphs helps characterize the limits of spectral behavior and often reveals optimal structures in network design or analysis. Overall, this project aims to contribute novel results to the growing body of work in matrix and spectral inequalities, with potential applications in various areas of mathematics and data science.