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

Let Phi =(G, phi) be a connected complex unit gain graph (T-gain graph) of nvertices with largest vertex degree Delta, adjacency matrix A(Phi), and degree matrix D(Phi). Let m(alpha)(Phi, lambda) be the multiplicity of.as an eigenvalue of A(alpha)(Phi) := alpha D(Phi) +(1 - alpha) A(Phi), for alpha is an element of[0, 1). In this article, we establish that m(alpha)(Phi, lambda) <= (Delta-2)n+2/Delta-1and characterize the sharpness. Then, we obtain some lower bounds for the rank r(Phi) in terms of n and Lambda including r(Phi) >= n-2/Delta-1and 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 Phi 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(Phi) in terms of the number of edges and characterize the sharpness. Result about m(alpha)(Phi, lambda) extends the corresponding known result for undirected graphs and simplifies the existing proof, and other bounds of r(Phi) obtained in this article work better than the bounds given elsewhere. (c) 2023 Elsevier B.V. All rights reserved.