In this paper, we aim to address the open questions raised in various recent papers regarding characterization of circulant graphs with three or four distinct eigenvalues in their spectra. Our focus is on providing characterizations and constructing classes of graphs falling under this specific category. We present a characterization of circulant graphs with prime number order and unitary Cayley graphs with arbitrary order, both of which possess spectra displaying three or four distinct eigenvalues. Various constructions of circulant graphs with composite orders are provided whose spectra consist of four distinct eigenvalues. These constructions primarily utilize specific subgraphs of circulant graphs that already possess two or three eigenvalues in their spectra, employing graph operations like the tensor product, the union, and the complement. Finally, we characterize the iterated line graphs of unitary Cayley graphs whose spectra contain three or four distinct eigenvalues, and we show their non-circulant nature.

翻译：摘要：本文旨在解决近期多篇论文中提出的关于刻画谱中具有三个或四个不同特征值的循环图这一开放问题。我们重点提供此类图的刻画方法并构造相关图类。我们给出了素数阶循环图与任意阶酉Cayley图的特征刻画，这两类图的谱均呈现三个或四个不同特征值。此外，我们提供了多种复合阶循环图的构造方法，其谱由四个不同特征值组成。这些构造主要利用谱中已存在两个或三个特征值的循环图子图，并采用张量积、并集和补图等图运算。最后，我们刻画了谱中包含三个或四个不同特征值的酉Cayley图迭代线图，并证明其非循环图性质。