A novel graph structure modeling a linear time invariant system following the Laplacian dynamics is proposed. This structure is based on the combination of two graphs and each of them has exactly two vertices with the same degree. It is shown that if the difference of vertex numbers of the two graphs is at most one then the combined graph is controllable by a single controller. A sufficient condition concerning the vertex selection for receiving control signals to guarantee the controllability of the combined graph is also proposed. This structure features a small sum of the maximum vertex degree and the diameter compared to some well-known controllable graphs. Numerical examples are provided to illustrate our work.
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