Abstract
Controllable graphs are connected graphs in which all eigenvalues are mutually distinct and main. In this work, a new method of characterizing the controllability of graphs with diameter n−2 is presented. A necessary and sufficient condition determining non-main eigenvalue of graphs with diameter n−2 is obtained, and the controllability of two kinds of graphs with diameter n−2 is characterized. Besides, the visualization representation of statistical results of controllable graphs is presented, and they show that the proportion of controllable graphs among the graphs with diameter n−2 is stablely at 15%, which partly verifies a conjecture proposed by Stanić.
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