On bounds of Aα-eigenvalue multiplicity and the rank of a complex unit gain graph

Abstract

Let Φ=(G,φ) be a connected complex unit gain graph (T-gain graph) of n vertices with largest vertex degree Δ, adjacency matrix A(Φ), and degree matrix D(Φ). Let mα(Φ,λ) be the multiplicity of λ as an eigenvalue of Aα(Φ):=αD(Φ)+(1−α)A(Φ), for α∈[0,1). In this article, we establish that mα(Φ,λ)≤(Δ−2)n+2Δ−1 and characterize the sharpness. Then, we obtain some lower bounds for the rank r(Φ) in terms of n and Δ including r(Φ)≥n−2Δ−1 and characterize their sharpness. Besides, we introduce zero-2-walk gain graphs and study their properties. It is shown that a zero-2-walk gain graph is always regular. Furthermore, we prove that Φ has exactly two distinct eigenvalues with equal magnitude if and only if it is a zero-2-walk gain graph. Using this, we establish a lower bound of r(Φ) in terms of the number of edges and characterize the sharpness. Result about mα(Φ,λ) extends the corresponding known result for undirected graphs and simplifies the existing proof, and other bounds of r(Φ) obtained in this article work better than the bounds given elsewhere.