Can you hide in plain sight?
Katherine Perry, assistant professor of mathematics at Soka University of America, explores if mathematics can.
Katherine Perry is a mathematician specializing in graph theory, design theory, and combinatorics. She is especially interested in breaking apart graphs into smaller subgraphs with special properties. She has a BA in mathematics from Scripps College and received both her master’s and PhD in mathematics from Auburn University. Prior to joining Soka University of America, she was a postdoc at the University of Denver.
In her classrooms, she aims to create an atmosphere of discovery through a variety of techniques designed to activate students’ abilities to think critically. Rather than simply explaining the material, she challenges students to see a problem and organically develop a process to solve it themselves, with confidence at efficiency. She teaches a variety of math courses, but some of her favorites are discrete mathematics, graph theory, mathematics for liberal arts, and linear algebra.
The Mathematics of Hiding in Plain Sight
How do we tell things apart? This question might seem simple, but in mathematics, it has profound implications. My research focuses on distinguishing numbers and paint cost in graph theory—how we can uniquely label elements of a network so that no two are confused.
Consider a perfect k-ary tree—a structured network where each node branches into k others. In my recent work, we introduce the concept of the paint cost spectrum, which measures the minimum effort needed to uniquely label such trees. This concept is not just theoretical—it has applications in cybersecurity, chemistry, and even robotics.
For example, in cybersecurity, networks must be structured so that an intruder can’t disguise themselves within the system. In chemistry, molecules can be modeled as networks, and distinguishing numbers help identify unique structures among similar compounds.
A key part of my research investigates how efficiently we can distinguish a network with the fewest possible labels. While some graphs are easily distinguished, others require surprisingly many labels, revealing deeper structural patterns. Understanding these properties helps optimize algorithms that identify vulnerabilities in networks or improve data organization.
This work connects to a larger question in mathematics: How do we quantify symmetry and uniqueness? By understanding these patterns, we can improve algorithms that identify vulnerabilities in networks or optimize search strategies in large datasets.
Mathematics often reveals hidden complexity in seemingly simple problems. By exploring the symmetry, we gain insights that shape the way we understand and navigate the interconnected world around us.
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