On quantum graph theory: non-commutative graph theory

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.