Bounds on the nullity, the H-rank and the Hermitian energy of a mixed graph. Issue 13 (3rd October 2021)

• Record Type: Journal Article
• Authors: Wei, Wei; Li, Shuchao; Ma, Hongping
• Is Part Of: Linear & multilinear algebra. Volume 69:Issue 13(2021)
• Page Start: 2469
• Page End: 2490
• Publication Date: 2021-10-03
• Subjects: Mixed graphs -- nullity -- Hermitian energy -- switching equivalence; 05C50 -- 05C35 -- 05C75; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2019.1679702
• Languages: English
• ISSNs: 0308-1087

Abstract

Abstract : Given an n -vertex graph G with maximum degree Δ, the mixed graph D G is constructed from G by orienting some of its edges, where G is called the underlying graph of D G . Let r H ( D G ) be the H -rank of D G and let α ( G ) be the independence number of G . In this paper, we firstly determine the maximum nullity of n -vertex mixed graphs with maximum degree Δ. The corresponding extremal graphs are identified. We secondly establish upper and lower bounds on r H ( D G ) + α ( G ), r H ( D G ) − α ( G ), r H ( D G ) / α ( G ), respectively. Finally, we characterize the relationship between the Hermitian energy and the H -rank of a mixed graph.