Some mixed graphs with H-rank 4, 6 or 8

Jinling Yang, Ligong Wang, Xiuwen Yang

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

The H-rank of a mixed graph Gα is defined to be the rank of its Hermitian adjacency matrix H(Gα). If Gα is switching equivalent to a mixed graph (Gα)′, and two vertices u, v of Gα have exactly the same neighborhood in (Gα)′, then u and v are said to be twins. The twin reduction graph TGα of Gα is a mixed graph whose vertices are the equivalence classes, and [u][v]∈E(TGα) if uv∈E((Gα)′), where [u] denotes the equivalence class containing the vertex u. In this paper, we give the upper (resp., lower) bound of the number of vertices of the twin reduction graphs of connected mixed bipartite graphs, and characterize all twin reduction graphs of the connected mixed bipartite graphs with H-rank 4 (resp., 6 or 8). Then, we characterize all connected mixed graphs with H-rank 4 (resp., 6 or 8) among all mixed graphs containing induced mixed odd cycles whose lengths are no less than 5 (resp., 7 or 9).

Original languageEnglish
Pages (from-to)678-693
Number of pages16
JournalJournal of Combinatorial Optimization
Volume41
Issue number3
DOIs
StatePublished - Apr 2021

Keywords

  • H-rank
  • Mixed bipartite graph
  • Switching equivalence

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