TL;DR
This paper reviews various methods for constructing quantum graphs, classifies certain quantum graphs, and introduces examples including a non-isomorphic quantum graph, advancing the understanding of quantum graph structures.
Contribution
It classifies all undirected quantum graphs on the quantum space M_2 and introduces new quantum graphs via cocycle deformations and non-isomorphic examples.
Findings
Classification of all undirected quantum graphs on M_2
Construction of quantum graphs via 2-cocycle deformations
Example of a quantum graph not quantum isomorphic to any classical graph
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 . 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.
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