Murray State Theses and Dissertations

Abstract

This thesis introduces and studies the notion of a strong neighborhood-prime labeling, a strengthening of the neighborhood-prime labeling where the label 1 can be assigned to any vertex in a graph. We prove that several graph families—including paths, cycles (excluding those congruent to 2 modulo 4), caterpillars, helm graphs, closed helm graphs, gear graphs, and graphs with universal vertices—admit such labelings, and also provide results to more general classes of graphs. We extend this new labeling concept to the Gaussian integers using a spiral order- ing on Z[i] and define a Gaussian analogue of strongly neighborhood-primeness. To support this extension, we verify a computational analogue of Bertrand’s Postulate, showing that a Gaussian prime exists between the nth and 2nth Gaussian integers of the spiral ordering for large n. Our results demonstrate that strongly neighborhood- prime behavior extends naturally to Z[i] for many graph families, enriching the theory of vertex labelings in both integer and complex settings.

Year manuscript completed

2025

Year degree awarded

2025

Author's Keywords

Graph Labeling, Graph Theory, Number Theory

Degree Awarded

Master of Science

Department

Mathematics & Statistics

College/School

Jesse D. Jones College of Science, Engineering and Technology

Thesis Advisor

Elizabeth Donovan

Committee Member

Kenny Barrese

Committee Member

Lesley Wiglesworth

Document Type

Thesis

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