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

TL;DR

This paper investigates the eigenvalue multiplicities and rank bounds of complex unit gain graphs, extending known results and providing new bounds based on graph degree and size.

Contribution

It establishes upper bounds for eigenvalue multiplicities and the rank of adjacency matrices in complex unit gain graphs, extending previous results and characterizing cases of equality.

Findings

Eigenvalue multiplicity bound: m_α(Φ, λ) ≤ ((Δ-2)n+2)/(Δ-1)

New bounds for the rank of A(Φ) in terms of graph parameters

Characterization of graph classes where bounds are tight

Abstract

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].