On quantum graph theory: non-commutative graph theory

Abu Omar, M. (2026) On quantum graph theory: non-commutative graph theory. MPhil(R) thesis, University of Glasgow.

Abstract

This research paper aims to introduce quantum graphs to newcomers. We adopt a clear and concise linear algebraic approach to simplify the concepts of quantum graphs. Our main perspective is viewing a quantum graph as a quantum adjacency matrix operator on a finite-dimensional C*-algebra (which can simply be thought of as a direct sum of matrix algebras), with the inner product defined by choosing a faithful positive linear functional on each matrix algebra summand of the direct sum. Our main result presents practical formulae for identifying isomorphic single-edged quantum graphs. An immediate corollary to this is that there is an infinite number of isomorphisms for a single-edged quantum graph on Mₙ(C) for n > 2, which is surprising since all single-edged quantum graphs are isomorphic to one another for n = 2.

• Item Type: Thesis (MPhil(R))
• Qualification Level: Masters
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering
• Supervisor's Name: Voigt, Professor Christian
• Date of Award: 2026
• Depositing User: Theses Team
• Unique ID: glathesis:2026-85721
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 28 Jan 2026 11:16
• Last Modified: 28 Jan 2026 11:16
• Thesis DOI: 10.5525/gla.thesis.85721
• URI: https://theses.gla.ac.uk/id/eprint/85721