Xi, Weige; Wang, Ligong The \(A_\alpha\) spectral radius and maximum outdegree of irregular digraphs. (English) Zbl 1506.05131 Discrete Optim. 38, Article ID 100592, 13 p. (2020). Summary: Let \(G\) be a digraph with adjacency matrix \(A(G)\). Let \(D(G)\) be the diagonal matrix with outdegrees of vertices of \(G\). In this paper, we study the convex linear combinations of \(A(G)\) and \(D(G)\), defined as \[ A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G),\quad 0\leq\alpha\leq 1. \] The largest modulus of the eigenvalues of \(A_\alpha(G)\), is called the \(A_\alpha\) spectral radius of \(G\), denoted by \(\lambda_\alpha(G)\). We establish some lower bounds on \(\Delta^+-\lambda_\alpha(G)\) for strongly connected irregular digraphs with given maximum outdegree and some other parameters, where \(\Delta^+\) is the maximum vertex outdegree of \(G\). Cited in 1 ReviewCited in 4 Documents MSC: 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 15A18 Eigenvalues, singular values, and eigenvectors 05C20 Directed graphs (digraphs), tournaments Keywords:strongly connected graph; \( A_\alpha\) spectral radius; irregular digraph PDFBibTeX XMLCite \textit{W. Xi} and \textit{L. Wang}, Discrete Optim. 38, Article ID 100592, 13 p. (2020; Zbl 1506.05131) Full Text: DOI References: [1] Hong, W. X.; You, L. H., Spectral radius and signless Laplacian spectral radius of strongly connected digraphs, Linear Algebra Appl., 457, 93-113 (2014) · Zbl 1295.05115 [2] Jin, Y. L.; Zhang, X. D., On the spectral radius of simple digraphs with prescribed number of arcs, Discrete Math., 338, 1555-1564 (2015) · Zbl 1311.05075 [3] Lin, H. Q.; Drury, S. W., The maximum Perron roots of digraphs with some given parameters, Discrete Math., 313, 2607-2613 (2013) · Zbl 1281.05070 [4] Xi, W. G.; So, W.; Wang, L. 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