  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].
