Bounds of nullity for complex unit gain graphs

A complex unit gain graph, or $T$-gain graph, is a triple $\Phi =(G,T,\phi )$ comprised of a simple graph G as the underlying graph of Φ, the set of unit complex numbers $T=\{z\in C:|z|=1\}$, and a gain function $\phi :\overrightarrow{E}\to T$ with the property that $\phi \left(e_{ij}\right)=\phi {\left(e_{ji}\right)}^{-1}$. A cactus graph is a connected graph in which any two cycles have at most one vertex in common.

In this paper, we firstly show that there does not exist a complex unit gain graph with nullity $n\left(G\right)-2m\left(G\right)+2c\left(G\right)-1$, where $n\left(G\right)$, $m\left(G\right)$ and $c\left(G\right)$ are the order, matching number, and cyclomatic number of G. Next, we provide a lower bound on the nullity for connected complex unit gain graphs and an upper bound on the nullity for complex unit gain bipartite graphs. Finally, we characterize all non-singular complex unit gain bipartite cactus graphs, which generalizes a result in Wong et al. (2022) [30].