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Let ∞∼-digraph be a generalized strongly connected ∞-digraph and let θ∼1-digraph and θ∼2-digraph be two kinds of generalized strongly connected θ-digraphs. In this paper, we characterize the extremal digraphs which achieve the maximum and minimum spectral radius among all Administrator in ∞∼-digraphs and θ∼1-digraphs. At the same time, we also determine the extremal digraph which achieves the maximum spectral radius among all Administrator in θ∼2-digraphs. Furthermore, we show that any ∞∼(a,b,c)-digraph is determined by the spectrum.