Algebraic connectedness and bipartiteness of quantum graphs

Abstract

Connectedness and bipartiteness are basic properties of classical graphs, and the purpose of this paper is to investigate the case of quantum graphs. We introduce the notion of connectedness and bipartiteness of quantum graphs in terms of graph homomorphisms. This paper shows that regular tracial quantum graphs have the same algebraic characterization of connectedness and bipartiteness as classical graphs. We also prove the equivalence between bipartiteness and two-colorability of quantum graphs by comparing two notions of graph homomorphisms respecting adjacency matrices or edge spaces. In particular, all kinds of quantum two-colorability are mutually equivalent for regular connected tracial quantum graphs.

arXiv comment31 pages

References in corpus (7)

• Zero-error communication via quantum channels, non-commutative graphs and a quantum Lovasz theta function
• Categorical formulation of quantum algebras
• A compositional approach to quantum functions
• Bigalois extensions and the graph isomorphism game
• The quantum-to-classical graph homomorphism game
• Some examples of quantum graphs
• Classification of Quantum Graphs on $M_2$ and their Quantum Automorphism Groups