#### Date of Thesis

Spring 2023

#### Description

Stable polynomials, in the context of this research, are two-variable polynomials like $p(z_1,z_2) = 2 - z_1 - z_2$ that are guaranteed to be non-zero if both input variables have an absolute value less than one in the complex plane. Stable polynomials are used in a variety of mathematical fields, thus finding ways to construct stable polynomials is valuable. An important property of these polynomials is whether they have boundary zeros, which are points in the complex plane where the polynomial equals zero and both variables have an absolute value of 1. Overall, it is challenging to find stable polynomials with boundary zeros.

Our research focuses on a construction method that goes from graphs (collections of vertices connected by edges) to matrices (2-dimensional arrays of numbers) to stable polynomials. To better understand the stable polynomials obtained by this construction, we are motivated to discover the relationship between the initial graph and the final stable polynomial. The first collection of results shows that whether the first vertex and the last vertex are connected by some path in the graph determines whether the stable polynomial constructed has a boundary zero, and all constructed polynomials must have a specific boundary zero if they have one. The second collection of results shows that if the initial graph has one of five patterns, then the constructed polynomial is guaranteed to have a specific additional boundary zero. Our results enable the construction of stable polynomials with specific boundary zeros and the identification of boundary zeros based on given stable polynomial properties.

#### Keywords

Stable polynomial, Function theory, Graph theory, Complex Analysis, Adjacancy matrix

#### Access Type

Honors Thesis

#### Degree Type

Bachelor of Science

#### Major

Mathematics

#### Second Major

Computer Science

#### First Advisor

Kelly Bickel

#### Second Advisor

Pamela Gorkin

#### Third Advisor

Emily Dryden

#### Recommended Citation

Hong, Yang, "Graphs, Adjacency Matrices, and Corresponding Functions" (2023). *Honors Theses*. 653.

https://digitalcommons.bucknell.edu/honors_theses/653