Junichiro Matsuda, Department of Pure Mathematics, University of Waterloo
Quantum graphs violate the classical characterization of the existence of d-regular graphs.
Quantum graphs are a non-commutative analogue of classical graphs that replace the function algebra on vertices with C*-algebras. It is known that classical simple d-regular graphs on n points exist if and only if dn is even. This is false for quantum graphs in both directions. We provide a necessary condition on the number of quantum edges between quantum vertices (matrix summands) to make it a d-regular quantum graph. Using this technique, we also describe 1-regular or 2-regular quantum graphs on general tracial quantum sets. 1-regular quantum graphs have quantum edges only between summands of the same size. Centrally connected $2$-regular quantum graphs are classified into 8 families by their central skeleton. This is a joint work with Matthew Kennedy and Larissa Kroell.
Hybrid - QNC 1507, https://uwaterloo.zoom.us/j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09