Some examples of quantum graphs

Abstract

We summarize different approaches to the theory of quantum graphs and provide several ways to construct concrete examples. First, we classify all undirected quantum graphs on the quantum space $M_2$. Secondly, we apply the theory of 2-cocycle deformations to Cayley graphs of abelian groups. This defines a twisting procedure that produces a quantum graph, which is quantum isomorphic to the original one. For instance, we define the anticommutative hypercube graphs. Thirdly, we construct an example of a quantum graph, which is not quantum isomorphic to any classical graph.

43 pages; added §5.4, some additional minor corrections

References in corpus (3)

• Classification of Quantum Graphs on $M_2$ and their Quantum Automorphism Groups
• Quantum symmetries of Cayley graphs of abelian groups
• Presentations of projective quantum groups

Cited by in corpus (5)

• Quantum graphs: different perspectives, homomorphisms and quantum automorphisms
• Classification of Quantum Graphs on $M_2$ and their Quantum Automorphism Groups
• Quantum graphs, subfactors and tensor categories I
• Complete Classification of Directed Quantum Graphs on M2
• Algebraic connectedness and bipartiteness of quantum graphs