On the multiplicity of $Aα$-eigenvalues and the rank of complex unit gain graphs

Let $Φ=(G, φ)$ be a connected complex unit gain graph ($\mathbb{T}$-gain graph) on a simple graph $G$ with $n$ vertices and maximum vertex degree $Δ$. The associated adjacency matrix and degree matrix are denoted by $A(Φ)$ and $D(Φ)$, respectively. Let $m_α(Φ,λ)$ be the multiplicity of $λ$ as an eigenvalue of $A_α(Φ) :=αD(Φ)+(1-α)A(Φ)$, for $α\in[0,1)$. In this article, we establish that $m_α(Φ, λ)\leq \frac{(Δ-2)n+2}{Δ-1}$, and characterize the classes of graphs for which the equality hold. Furthermore, we establish a couple of bounds for the rank of $A(Φ)$ 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 $r(Φ)$ than the bounds known in [8].

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