We define k-dimensional digraphs and initiate a study of their spectral theory. The k-dimensional digraphs can be viewed as generating graphs for small categories called k-graphs. Guided by geometric insight, we obtain several new series of k-graphs using cube complexes covered by Cartesian products of trees, for k≥2. These k-graphs cannot be presented as virtual products and constitute novel models of such small categories. The constructions yield rank-k Cuntz-Krieger algebras for all k≥2. We introduce Ramanujan k-graphs satisfying optimal spectral gap property and show explicitly how to construct the underlying k-digraphs.
17 сен 2024
