Quantum symmetry in multigraphs (part I)

TL;DR

This paper introduces and explores various notions of quantum symmetry in multigraphs, showing how they relate to existing concepts and identifying conditions under which genuine quantum symmetry exists.

Contribution

It defines new notions of quantum symmetry for multigraphs and demonstrates their relation to existing theories, highlighting conditions for genuine quantum symmetry.

Findings

All notions reduce to known quantum symmetries in simple graphs.

Multigraphs with multiple edges between at least two pairs of vertices have genuine quantum symmetry.

The study extends quantum symmetry concepts from simple graphs to multigraphs.

Abstract

We introduce various notions of quantum symmetry in a directed or undirected multigraph with no isolated vertex and explore relations among them. If the multigraph is single edged (that is, a simple graph where loops are allowed), all our notions of quantum symmetry reduce to already existing notions of quantum symmetry provided by Bichon and Banica. Our constructions also show that any multigraph with at least two pairs of vertices with multiple edges among them possesses genuine quantum symmetry.