Constructing a controllable graph under edge constraints

This paper is concerned with the problem of constructing a controllable graph subject to some practical edge constraints. Specifically, suppose the total amount of vertices and the upper bounds on the graph diameter or on the vertex degree are given. We consider the problem of exploring a class of feasible graphs that satisfy the constraints. Using the hybrid of a path graph and an antiregular graph we propose a simple and systematic method to generate a class of controllable graphs whose diameters or degrees cover the full possible ranges. The method to select the control vector to ensure the controllability of the combined graph is also proposed. Numerical examples are provided to demonstrate our results.