Quantum Mycielskians: symmetries, twin vertices and distinguishing labelings

TL;DR

This paper explores a quantum extension of the Mycielski construction for graphs, analyzing quantum symmetries, twin vertices, and a new quantum distinguishing number, revealing how classical concepts translate into the quantum domain.

Contribution

It introduces a quantum analogue of graphs with twin vertices and a quantum extension of the distinguishing number, advancing the understanding of quantum graph symmetries.

Findings

Quantum symmetries of Mycielskians relate to underlying quantum graphs

Quantum twin vertices generalize classical twin vertices with consistent reduction

Quantum distinguishing number behaves predictably under the quantum Mycielski construction

Abstract

We investigate a quantum generalization of the Mycielski construction for quantum graphs. In particular, we analyze the quantum symmetries of quantum Mycielskians and their relation to the symmetries of the underlying quantum graphs. We introduce a quantum analogue of graphs with twin vertices and show that, for classical graphs with a small number of vertices, this notion reduces to the classical one. Finally, we propose a quantum extension of the distinguishing number and examine its behavior under the quantum Mycielski construction.