Research Assistant, The Metric Dimension and

3 weeks ago

Winnipeg, Canada University of Winnipeg Full time

Research Assistants are normally students, in a University of Winnipeg undergraduate program, employed to assist with tasks related to research. Work is performed within defined guidelines and under supervision of a more senior researcher or faculty member.

**Position Overview**:
A graph or hypergraph consists of a set of points called vertices, and a set of subsets of this vertex set called edges. Graphs are used to model many types of networks, such as transportation systems, the link structure of websites, and data organization networks. We will study a useful graph parameter, called the metric dimension. The metric dimension of a graph can be equated to the minimum number of distance-detecting devices required to locate an intruder in a network. It is natural to ask if the number of detecting devices that are required could be reduced if additional links are added to the network. This prompts the question by how much the metric dimension of a graph can be reduced by adding edges to the graph, which is the motivation for studying the so-called threshold dimension of a graph. The problem of determining the metric dimension and the threshold dimension of a graph is NP hard, and so graph theorists focus on determining this parameter for different classes of graphs. We focus on determining the metric and threshold dimension of Cayley graphs and hypergraphs, which have a high level of symmetry and are constructed algebraically.

The research team will consist of myself and one undergraduate student researcher. I will supervise and direct the project, but the goal is for the student to achieve some new results independently and co-author a research paper. We will begin by reviewing the relevant literature. We will draw from the techniques used by other researchers, and get a good handle on what has been done and what questions remain and are of interest to researchers in this area. We use algebraic techniques and properties of the underlying group to determine the metric dimension of Cayley graphs on groups, and an analogous approach will be useful in determining the metric and threshold dimension of Cayley hypergraphs The algebraic construction of the Cayley hypergraphs will allow us to determine the distance between two vertices by an algebraic formula, which will be very advantageous for computing their metric dimension.

This project is a great opportunity for a senior undergraduate student to experience mathematical research and problem solving first-hand. Initially the student will receive background training. They will begin with a literature review, to master the common techniques for bounding the metric dimension and constructing resolving sets for well known graphs. The student will read about the recent work on the metric and threshold dimension of a graph, and about several classes of groups and how to construct their Cayley graphs and hypergraphs. Ideally the student will discover some new results on the metric and threshold dimension of these graphs and hypergraphs independently.

The student will have opportunities for professional development while helping to write up the new results we achieve for publication in a mathematical journal, using LaTeX or another math text editor. They may also have the opportunity to present the results of our research at an undergraduate mathematics conference or poster competition. The student will benefit academically through learning about the important interplay between groups and discrete structures such as hypergraphs. This project will reinforce and strengthen the skills needed to succeed in graduate level research.

The successful applicant will work approximately 30 hours per week, for 14 weeks, from May 15-August 18 2023, for a total of 420 working hours. The position entails directed readings and discussions, as well as independent study. The work can be completed remotely, or in an office space provided on campus. We will meet each day to discuss different aspects of the problem, and share our progress. We can meet via Zoom or in person.

**Duties**:

- Perform academic research, standard tests and experiments.
- Employee will be assigned directed readings and exercises to learn about the problem, and carry out a literature search for new publications in this area.
- Monitor and document research status and findings.
- Employee will document their progress and write up any new results they achieve using LaTeX or another math editor.
- Assist with duties related to the production of articles, publications, reports and presentations.
- Employee will assist with writing and editing a mathematical research article. They may present their results at an undergraduate math conference or poster competition.

**Qualifications**:

- Required Courses: MATH-1201 (Linear Algebra I), MATH-1401 (Discrete Math), MATH-1103 (Intro Calculus I), MATH-1104 (intro Calculus II).
- Computer programming skills are an asset.
- Knowledge of LaTeX or another math editor is an a

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