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

TL;DR

This paper characterizes complex unit gain bicyclic graphs with small ranks (2, 3, or 4), extending previous work on unicyclic graphs and their inertia properties.

Contribution

It provides a complete characterization of bicyclic graphs with small rank in the context of complex unit gain graphs, which was not previously known.

Findings

Identified all bicyclic graphs with rank 2, 3, or 4.

Extended prior results from unicyclic to bicyclic graphs.

Contributed to the understanding of the spectral properties of complex unit gain graphs.

Abstract

A $\mathbb{T}$-gain graph is a triple $\Phi=(G,\mathbb{T},\varphi)$ consisting of a graph $G=(V,E)$, the circle group $\mathbb{T}=\{z\in C: |z|=1\}$ and a gain function $\varphi:\overrightarrow{E}\rightarrow \mathbb{T}$ such that $\varphi(e_{ij})=\varphi(e_{ji})^{-1}=\overline{\varphi(e_{ji})}$. 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.