Minimum Laplacian controllability of graphs based on interconnecting two classes of threshold graphs

In this work we study the Laplacian controllability of a class of connected simple graphs. Consider two $k -$ vertex threshold graphs, each with its own unique repeated degree. Suppose the multiplicities of the repeated degrees in the first and second graphs are m 1 and m 2 respectively, where $m_{1}\geq$ m 2 . If the two threshold graphs are interconnected via a new edge, it is shown that the minimum number of controllers to render the resulting graph Laplacian controllable is $m_{1}-1$ if $m_{1} \gt $ m 2 , and is $2 m_{1}-3$ otherwise. The method to add this new edge and to connect these controllers to ensure the controllability is presented. Numerical examples are provided to illustrate our results.