On Almost Uniformly Locally Finite Graphs

We study locally finite graphs whose normalized adjacency operators belong to the C*-algebra T, the norm closure of the algebra of l1-bounded operators. We give sufficient and necessary conditions in terms of the degrees of adjacent vertices, prove stability results under finite edge perturbations and Cartesian products, and compute several examples: regular graphs, complete bipartite graphs, complete graphs with pendant leaves, subdivisions, and spherically homogeneous trees. These examples show that T-uniformity is a weak regularity condition which forbids vertices of large degree from being adjacent to too many vertices of much smaller degree.