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The maximum distance matrix of a strongly connected digraph is a symmetric matrix whose rows and columns are indexed the vertices, the entries of which correspond to the maximum directed distance between the vertices. In this paper, we determine the digraphs that uniquely minimize the largest eigenvalue of the maximum distance matrix in some classes of strongly connected digraphs, and the n -vertex strongly connected digraphs for which the maximum distance matrices have an eigenvalue with multiplicity n − 1.
Xu et al. (Tue,) studied this question.