TY - JOUR
T1 - The rank of a complex unit gain graph in terms of the rank of its underlying graph
AU - Lu, Yong
AU - Wang, Ligong
AU - Zhou, Qiannan
N1 - Publisher Copyright:
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2019/8/15
Y1 - 2019/8/15
N2 - Let Φ = (G, φ) be a complex unit gain graph (or T-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) - 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-θ(G)≤r(Φ)r(G)≤1+θ(G). All corresponding extremal graphs are characterized.
AB - Let Φ = (G, φ) be a complex unit gain graph (or T-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) - 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-θ(G)≤r(Φ)r(G)≤1+θ(G). All corresponding extremal graphs are characterized.
KW - 05C35
KW - 05C50
KW - Dimension of cycle space
KW - Rank of graphs
KW - T-gain graph
UR - http://www.scopus.com/inward/record.url?scp=85062692669&partnerID=8YFLogxK
U2 - 10.1007/s10878-019-00397-y
DO - 10.1007/s10878-019-00397-y
M3 - 文章
AN - SCOPUS:85062692669
SN - 1382-6905
VL - 38
SP - 570
EP - 588
JO - Journal of Combinatorial Optimization
JF - Journal of Combinatorial Optimization
IS - 2
ER -