Abstract We investigate the algebraic degree of circulant graphs, i.e. the dimension of the splitting field of the characteristic polynomial of the associated adjacency matrix over the rationals. Studying the algebraic degree of graphs seems more natural than characterizing graphs with integral spectra only. We prove that the algebraic degree of circulant graphs on n vertices is bounded above by φ ( n ) / 2 , where φ denotes Euler's totient function, and that the family of cycle graphs provides a family of maximum algebraic degree within the family of all circulant graphs. Moreover, we precisely determine the algebraic degree of circulant graphs on a prime number of vertices.

中文翻译：

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循环图谱的代数次数

摘要 我们研究了循环图的代数度，即相关邻接矩阵在有理数上的特征多项式的分裂域的维数。研究图的代数次数似乎比仅用积分谱表征图更自然。我们证明了 n 个顶点上的循环图的代数次数的上界为 φ ( n ) / 2 ，其中 φ 表示欧拉的整体函数，并且循环图族提供了所有循环图族内的最大代数次数族图表。此外，我们精确地确定了素数顶点上循环图的代数次数。