Quantum Mycielskians: symmetries, twin vertices and distinguishing labelings

Abstract: We study the properties of a generalization of the Mycielski construction to quantum graphs. We discuss quantum symmetries of quantum Mycielskians and their relation to quantum symmetries of the original quantum graphs. We introduce a quantum generalization of the notion of graphs with twin vertices and check that this concept coincides with its classical version for classical graphs with few vertices. Finally, we propose the quantum generalization of the distinguishing number and study its behavior under the quantum Mycielski construction.