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

Jinling Yang (), Ligong Wang () and Xiuwen Yang ()
Additional contact information
Jinling Yang: Northwestern Polytechnical University
Ligong Wang: Northwestern Polytechnical University
Xiuwen Yang: Northwestern Polytechnical University

Journal of Combinatorial Optimization, 2021, vol. 41, issue 3, No 5, 678-693

Abstract: Abstract The H-rank of a mixed graph $$G^{\alpha }$$ G α is defined to be the rank of its Hermitian adjacency matrix $$H(G^{\alpha })$$ H ( G α ) . If $$ G^{\alpha } $$ G α is switching equivalent to a mixed graph $$(G^{\alpha })' $$ ( G α ) ′ , and two vertices u, v of $$G^{\alpha }$$ G α have exactly the same neighborhood in $$(G^{\alpha })'$$ ( G α ) ′ , then u and v are said to be twins. The twin reduction graph $$T_{G^{\alpha }}$$ T G α of $$G^{\alpha }$$ G α is a mixed graph whose vertices are the equivalence classes, and $$[u][v]\in E(T_{G^{\alpha }})$$ [ u ] [ v ] ∈ E ( T G α ) if $$uv\in E((G^{\alpha })')$$ u v ∈ E ( ( G α ) ′ ) , 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).

Keywords: H-rank; Switching equivalence; Mixed bipartite graph; 05C05 (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10878-021-00704-6 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jcomop:v:41:y:2021:i:3:d:10.1007_s10878-021-00704-6

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10878

DOI: 10.1007/s10878-021-00704-6

Access Statistics for this article

Journal of Combinatorial Optimization is currently edited by Thai, My T.

More articles in Journal of Combinatorial Optimization from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().