Abstract
Let G be a strongly connected digraph with distance matrix D(G) and let Tr(G) be the diagonal matrix with vertex transmissions of G. For any real ? ? [0, 1], define the matrix D?(G) as D?(G) = ?Tr(G) + (1-?)D(G). The D? spectral radius of G is the spectral radius of D?(G). In this paper, we first give some upper and lower bounds for the D? spectral radius of G and characterize the extremal digraphs. Moreover, for digraphs that are not transmission regular, we give a lower bound on the difference between the maximum vertex transmission and the D? spectral radius. Finally, we obtain the D? eigenvalues of the join of certain regular digraphs.
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