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

Abstract

Let $\Phi =(G,\phi )$ be a complex unit gain graph (or $\mathbb{T}$-gain graph) and $A(\Phi )$ be its adjacency matrix, where $G$ is called the underlying graph of $\Phi$. The rank of $\Phi$, denoted by $r(\Phi )$, is the rank of $A(\Phi )$. Denote by $\theta (G)=|E(G)|-|V(G)|+\omega (G)$ the dimension of cycle spaces of $G$, where $|E(G)|$, $|V(G)|$ and $\omega (G)$ 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 $r(\Phi )$ in terms of $r(G)$, that is, $r(G)-2\theta (G)\le r(\Phi )\le r(G)+2\theta (G)$, where $r(G)$ is the rank of $G$. As an application, we also prove that $1-\theta (G)\le \frac{r(\Phi )}{r(G)}\le 1+\theta (G)$. All corresponding extremal graphs are characterized.

17 pages. arXiv admin note: text overlap with arXiv:1612.05043

References in corpus (1)

• Skew-rank of an oriented graph in terms of the rank and dimension of cycle space of its underlying graph