I will discuss various definitions of quantum or noncommutative graphs that have appeared in the literature, along with motivating examples. One definition is due to Weaver, where examples arise from quantum channels and the study of quantum zero-error communication. This definition works for any von Neumann algebra, and is "spatial": an operator system satisfying a certain operator bimodule condition. Another definition, first due to Musto, Reutter, and Verdon, involves a generalisation of the concept of an adjacency matrix, coming from the study of (simple, undirected) graphs. Here we study finite-dimensional C*-algebras with a given faithful state; examples are perhaps less obvious. I will discuss generalisations of the latter framework when the state is not tracial, and discuss various notions of a "morphism" of the resulting objects
