The spectra of random mixed graphs

Dan Hu, Hajo Broersma*, Jiangyou Hou, Shenggui Zhang

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

1 Citation (Scopus)
44 Downloads (Pure)

Abstract

A mixed graph is a graph that can be obtained from a simple undirected graph by replacing some of the edges by arcs in precisely one of the two possible directions. The Hermitian adjacency matrix of a mixed graph G of order n is the n×n matrix H(G)=(hij), where hij=−hji=i (with i=−1) if there exists an arc from vi to vj (but no arc from vj to vi), hij=hji=1 if there exists an edge (and no arcs) between vi and vj, and hij=hji=0 otherwise (if vi and vj are neither joined by an edge nor by an arc). Let λ1(G),λ2(G),…,λn(G) be eigenvalues of H(G). The k-th Hermitian spectral moment of G is defined as sk(H(G))=∑i=1nλik(G), where k≥0 is an integer. In this paper, we deal with the asymptotic behavior of the spectrum of the Hermitian adjacency matrix of random mixed graphs. We will present and prove a separation result between the largest and remaining eigenvalues of the Hermitian adjacency matrix, and as an application, we estimate the Hermitian spectral moments of random mixed graphs.

Original languageEnglish
Pages (from-to)320-338
Number of pages19
JournalLinear algebra and its applications
Volume653
DOIs
Publication statusPublished - 15 Nov 2022

Keywords

  • Hermitian spectral moment
  • Random Hermitian adjacency matrix
  • Random mixed graphs
  • Spectrum
  • UT-Hybrid-D

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