Further results on controllable graphs

Connected graphs whose eigenvalues are mutually distinct and main are called controllable graphs. In recent work their relevance in control theory is recognized, and a number of theoretical and computational results are obtained. In this paper, some criteria for non-controllability of graphs are considered, and certain constructions of controllable graphs are given. Controllable graphs whose index does not exceed a given constant (close to 2.0366) are limited as part of two specific families of trees, and controllable graphs with extremal diameter are discussed. Some computational results are presented, along with corresponding theoretical observations.