Graphs, Adjacency Matrices and Stable Polynomials
Author:Yang Hong ’23
Faculty Mentor(s):Kelly Bickel, Mathematics department
Our research concerns the interplay of undirected graphs and stable polynomials. Stable polynomials, which are polynomials with restricted zero sets, are used in a variety of mathematical fields. Here, a stable polynomial p is defined as a two-variable polynomial that satisfies p(z_1,z_2) ≠ 0 for any (z_1,z_2) in the unit disk. In this research, we use adjacency matrices of undirected graphs to construct stable polynomials and investigate the relationships between the shapes of the graphs and the zeros of the polynomials. In a variety of situations, we establish the existence and location of polynomial zeros on the boundary of the unit disk and characterize how the zero set of a stable polynomial could approach those boundary zeros. We also pose and examine conjectures about more generalized and complicated cases. Using our results, one can build polynomials with specific boundary zeros and identify the boundary zeros implied by given polynomial properties.