On the multiplicity of $A{\alpha}$-eigenvalues and the rank of complex unit gain graphs

Let $\Phi=(G, \varphi)$ be a connected complex unit gain graph ($\mathbb{T}$-gain graph) on a simple graph $G$ with $n$ vertices and maximum vertex degree $\Delta$. The associated adjacency matrix and degree matrix are denoted by $A(\Phi)$ and $D(\Phi)$, respectively. Let $m_{\alpha}(\Phi,\lambda)$ be the multiplicity of $\lambda$ as an eigenvalue of $A_{\alpha}(\Phi) :=\alpha D(\Phi)+(1-\alpha)A(\Phi)$, for $\alpha\in[0,1)$. In this article, we establish that $m_{\alpha}(\Phi, \lambda)\leq \frac{(\Delta-2)n+2}{\Delta-1}$, and characterize the classes of graphs for which the equality hold. Furthermore, we establish a couple of bounds for the rank of $A(\Phi)$ in terms of the maximum vertex degree and the number of vertices. One of the main results extends a result known for unweighted graphs and simplifies the proof in [15], and other results provide better bounds for $r(\Phi)$ than the bounds known in [8].