New results in vertex sedentariness

A vertex in a graph is said to be sedentary if a quantum state assigned to that vertex tends to stay on that vertex. Using the direct product, join, and blow-up operations, we construct new infinite families of graphs with sedentary vertices. We prove our main result, which states that a vertex with a twin is either sedentary or admits pretty good state transfer with its twin. In contrast, we provide infinite families of graphs containing vertices with no twins that are neither sedentary nor involved in pretty good state transfer. Using our main result, we completely characterize twin vertices that are sedentary and provide improved bounds on their sedentariness. This allows us to construct an infinite family of trees with maximum degree three that admit vertex sedentariness. We also determine all complete bipartite graphs and threshold graphs that contain sedentary vertices relative to the Laplacian matrix.