Zachary King, Undergraduate Researcher in Mathematics

Zachary King is an undergraduate student majoring in pure mathematics at the University of Central Oklahoma and is currently beginning his junior year. He has been working with Dr. Liz Lane-Harvard and Dr. Thomas Milligan on research on iterated line graphs since the second part of his freshman year, Spring 2018, picking up on a project originally conducted with former undergraduate student Brendan Balch, now a Ph.D. student at Colorado State University. For a given graph G, defined as a set of vertices together with a set of edges between pairs of vertices, we form L(G), the line graph of G, by creating a new vertex corresponding to each edge in the original graph and edges between these vertices if the original edges in G shared a vertex. When considering iterated line graphs, we are looking at G, L(G), its line graph L(L(G)), and so on, successively. Brendan originally looked at successive line graphs of star graphs and variations of those graphs. Zach has extended these results by focusing on the structure of a subgraph common to line graphs, namely where two regular induced subgraphs are connected by a bi-regular graph. Zachary has received funding from STLR and presented a poster at Oklahoma Research Day in March. He also presented his research at the Oklahoma-Arkansas Sectional Meeting of the Mathematical Association of America in Tahlequah, OK on March 29th, taking third place among the student presentations. Furthermore, Zachary presented at the Central States Math Undergraduate Research Conference at Kansas State University in Manhattan, KS on April 12th. Zachary is finalizing a paper with his results and already working on a second paper over related results. He has also started working on another research project with Dr. R. Scott Williams involving looking at Pascal’s triangle modulo m>2 with a goal of understanding the relationship between its structure and the prime factors of m.

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