The Aα spectral radius and maximum outdegree of irregular digraphs

Abstract 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 α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) , 0 ≤ α ≤ 1 . The largest modulus of the eigenvalues of A α ( G ) , is called the A α spectral radius of G , denoted by λ α ( G ) . We establish some lower bounds on Δ + − λ α ( G ) for strongly connected irregular digraphs with given maximum outdegree and some other parameters, where Δ + is the maximum vertex outdegree of G .

[1]  V. Nikiforov,et al.  On the α -index of graphs with pendent paths , 2017, Linear Algebra and its Applications.

[2]  D. Jiang,et al.  The signless Laplacian spectral radius of k-connected irregular graphs , 2018, Linear Algebra and its Applications.

[3]  Hao Li,et al.  On the signless Laplacian spectral radius of irregular graphs , 2013 .

[4]  Bolian Liu,et al.  The A spectral radius characterization of some digraphs , 2019, Linear Algebra and its Applications.

[5]  D. Cvetkovic,et al.  An Introduction to the Theory of Graph Spectra: References , 2009 .

[6]  Jie Xue,et al.  On the A α -spectral radius of a graph , 2018, Linear Algebra and its Applications.

[7]  W. Shiu,et al.  The signless Laplacian spectral radius of k-connected irregular graphs , 2017 .

[8]  Jixiang Meng,et al.  The Aα-spectral radius of trees and unicyclic graphs with given degree sequence , 2019, Appl. Math. Comput..

[9]  Ya-Lei Jin,et al.  On the spectral radius of simple digraphs with prescribed number of arcs , 2015, Discret. Math..

[10]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[11]  Sebastian M. Cioaba,et al.  Extreme eigenvalues of nonregular graphs , 2007, J. Comb. Theory, Ser. B.

[13]  Xing Huang,et al.  A note on the A-spectral radius of graphs , 2018, Linear Algebra and its Applications.

[14]  Huiqiu Lin,et al.  The maximum Perron roots of digraphs with some given parameters , 2013, Discret. Math..

[16]  Xiaogang Liu,et al.  On the A-characteristic polynomial of a graph , 2018, Linear Algebra and its Applications.

[17]  L. You,et al.  Spectral radius and signless Laplacian spectral radius of strongly connected digraphs , 2014, 1401.7091.

[18]  V. Nikiforov,et al.  Merging the A- and Q-spectral theories , 2016, 1607.03015.

[19]  Xiaogang Liu,et al.  Graphs determined by their Aα-spectra , 2019, Discret. Math..

[20]  Ligong Wang,et al.  The (distance) signless Laplacian spectral radius of digraphs with given arc connectivity , 2019, Linear Algebra and its Applications.

[21]  Ligong Wang,et al.  On the signless Laplacian spectral radius of weighted digraphs , 2019, Discret. Optim..

[22]  Vladimir Nikiforov,et al.  A note on the positive semidefiniteness of Aα(G) , 2017 .

[23]  Sebastian M. Cioaba,et al.  The Spectral Radius and the Maximum Degree of Irregular Graphs , 2007, Electron. J. Comb..