Complex unit gain bicyclic graphs with rank 2, 3 or 4

Abstract

A $\mathbb{T}$-gain graph is a triple $\Phi =(G,\mathbb{T},\phi )$ consisting of a graph $G=(V,E)$, the circle group $\mathbb{T}=\{z\in C:|z|=1\}$ and a gain function $\phi :\overrightarrow{E}\to \mathbb{T}$ such that $\phi (e_{ij})=\phi (e_{ji}{)}^{-1}=\overline{\phi (e_{ji})}$. The rank of $\mathbb{T}$-gain graph $\Phi$, denoted by $r(\Phi )$, is the rank of the adjacency matrix of $\Phi$. In 2015, Yu, Qu and Tu [ G. H. Yu, H. Qu, J. H. Tu, Inertia of complex unit gain graphs, Appl. Math. Comput. 265(2015) 619--629 ] obtained some properties of inertia of a $\mathbb{T}$-gain graph. They characterized the $\mathbb{T}$-gain unicyclic graphs with small positive or negative index. Motivated by above, in this paper, we characterize the complex unit gain bicyclic graphs with rank 2, 3 or 4.

15 pages, 4 figures

References in corpus (2)

• Spectral Properties of Complex Unit Gain Graphs
• A survey on the skew energy of oriented graphs