ABSTRACT

A complex unit gain graph is a simple graph in which each orientation of an edge is given a complex number with modulus 1 and its inverse is assigned to the opposite orientation of the edge. In this article, first we establish bounds for the eigenvalues of the complex unit gain graphs. Then we study some of the properties of the adjacency matrix of a complex unit gain graph in connection with the characteristic and the permanental polynomials. Then we establish spectral properties of the adjacency matrices of complex unit gain graphs. In particular, using Perron–Frobenius theory, we establish a characterization for bipartite graphs in terms of the set of eigenvalues of a gain graph and the set of eigenvalues of the underlying graph. Also, we derive an equivalent condition on the gain so that the eigenvalues of the gain graph and the eigenvalues of the underlying graph are the same.

摘要

复数单位增益图是一个简单的图，其中边缘的每个方向都被赋予一个模数为 1 的复数，并且它的倒数被分配给边缘的相反方向。在本文中，首先我们为复单位增益图的特征值建立界限。然后我们结合特征和永久多项式研究了复单位增益图的邻接矩阵的一些性质。然后我们建立复单位增益图的邻接矩阵的谱特性。特别是，使用 Perron-Frobenius 理论，我们根据增益图的特征值集和基础图的特征值集建立了二部图的表征。还，