Quantum graphs are a non-commutative analogue of classical graphs related to operator algebras, quantum information, quantum groups, etc. In this talk, I will give a brief introduction to quantum graphs and talk about spectral characterizations of properties of quantum graphs. We introduce connectedness and bipartiteness of quantum graphs in terms of graph homomorphisms, and these properties have algebraic characterizations in the same way as classical cases. We also see the equivalence between bipartiteness and two-colorability of quantum graphs defined by two notions of graph homomorphisms: one respects adjacency matrices, and the other respects edge spaces.

This talk is based on arXiv:2310.09500.