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

2017/11/29 by Yong Lu, Lu, Yong, Ligong Wang +3
Computer Science · Mathematics · #05C35 #05C50 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Graph Labeling and Dimension Problems #Graph theory and applications

paper · pdf · doi:10.48550/arxiv.1711.11448

openalex publication_date 2017/11/29 · openalex created_date 2017/12/22 · openalex updated_date 2026/07/28

Abstract

Let Φ=(G, φ) be a complex unit gain graph (or \mathbbT-gain graph) and A(Φ) be its adjacency matrix, where G is called the underlying graph of Φ. The rank of Φ, denoted by r(Φ), is the rank of A(Φ). Denote by θ(G)=|E(G)|-|V(G)|+ω(G) the dimension of cycle spaces of G, where |E(G)|, |V(G)| and 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(Φ) in terms of r(G), that is, r(G)-2θ(G)≤ r(Φ)≤ r(G)+2θ(G), where r(G) is the rank of G. As an application, we also prove that 1-θ(G)≤(r(Φ))/(r(G))≤1+θ(G). All corresponding extremal graphs are characterized.

Citations

• Skew-rank of an oriented graph in terms of the rank and dimension of cycle space of its underlying graph
• Relation between the skew-rank of an oriented graph and the independence number of its underlying graph

Related

• Bounds for the rank of a complex unit gain graph in terms of the independence number
• The rank of a complex unit gain graph in terms of the matching number
• On the multiplicity of Aα-eigenvalues and the rank of complex unit gain graphs
• Bounds and extremal graphs for the energy of complex unit gain graphs
• Complex unit gain bicyclic graphs with rank 2, 3 or 4