The rank of a complex unit gain graph in terms of the rank of its underlying graph

Yong Lu (), Ligong Wang () and Qiannan Zhou ()
Additional contact information
Yong Lu: Jiangsu Normal University
Ligong Wang: Northwestern Polytechnical University
Qiannan Zhou: Northwestern Polytechnical University

Journal of Combinatorial Optimization, 2019, vol. 38, issue 2, No 13, 570-588

Abstract: Abstract Let $$\Phi =(G, \varphi )$$ Φ = ( G , φ ) be a complex unit gain graph (or $$\mathbb {T}$$ T -gain graph) and $$A(\Phi )$$ A ( Φ ) be its adjacency matrix, where G is called the underlying graph of $$\Phi $$ Φ . The rank of $$\Phi $$ Φ , denoted by $$r(\Phi )$$ r ( Φ ) , is the rank of $$A(\Phi )$$ A ( Φ ) . Denote by $$\theta (G)=|E(G)|-|V(G)|+\omega (G)$$ θ ( G ) = | E ( G ) | - | V ( G ) | + ω ( G ) the dimension of cycle spaces of G, where |E(G)|, |V(G)| and $$\omega (G)$$ ω ( G ) are the number of edges, the number of vertices and the number of connected components of G, respectively. In this paper, we investigate bounds for $$r(\Phi )$$ r ( Φ ) in terms of r(G), that is, $$r(G)-2\theta (G)\le r(\Phi )\le r(G)+2\theta (G)$$ r ( G ) - 2 θ ( G ) ≤ r ( Φ ) ≤ r ( G ) + 2 θ ( G ) , where r(G) is the rank of G. As an application, we also prove that $$1-\theta (G)\le \frac{r(\Phi )}{r(G)}\le 1+\theta (G)$$ 1 - θ ( G ) ≤ r ( Φ ) r ( G ) ≤ 1 + θ ( G ) . All corresponding extremal graphs are characterized.

Keywords: $$\mathbb {T}$$ T -gain graph; Rank of graphs; Dimension of cycle space; 05C35; 05C50 (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://link.springer.com/10.1007/s10878-019-00397-y 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:38:y:2019:i:2:d:10.1007_s10878-019-00397-y

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

DOI: 10.1007/s10878-019-00397-y

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 ().