Sharp Upper Bounds for the Largest Laplacian Eigenvalue of Mixed Graphs

Abstract

The paper presents two sharp upper bounds for the largest Laplacian eigenvalue of mixed graphs in terms of the degrees and the average $2$-degrees, which improve and generalize the main results of Zhang and Li [Linear Algebra Appl.$353(2002)11-20]$,Pan (Linear Algebra Appl.$355(2002)287-295]$,respectively. Moreover, we also characterize some extreme graphs which attain these upper bounds. In last, some examples show that our bounds are improvement on some known bounds in some cases.