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

Abstract

Let $\Phi =(G,\phi )$ 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 $m_{\alpha }(\Phi ,\lambda )\le \frac{(\Delta -2)n+2}{\Delta -1}$, 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].