Abstract
Let [Formula: see text] be a digraph of order [Formula: see text] and let [Formula: see text] be the adjacency matrix of [Formula: see text] Let Deg[Formula: see text] be the diagonal matrix of vertex out-degrees of [Formula: see text] For any real [Formula: see text] the generalized adjacency matrix [Formula: see text] of the digraph [Formula: see text] is defined as [Formula: see text] This matrix generalizes the spectral theories of the adjacency matrix and the signless Laplacian matrix of [Formula: see text]. In this paper, we find [Formula: see text]-spectrum of the joined union of digraphs in terms of spectrum of adjacency matrices of its components and the eigenvalues of an auxiliary matrix determined by the joined union. We determine the [Formula: see text]-spectrum of join of two regular digraphs and the join of a regular digraph with the union of two regular digraphs of distinct degrees. As applications, we obtain the [Formula: see text]-spectrum of various families of unsymmetric digraphs.
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