带圈的稀疏正则图的谱
Spectra of sparse regular graphs with loops
摘要：
我们推导出精确的方程，确定无向和有向稀疏连接的正则图包含任意长度的循环的频谱。我们的研究结果的网络的结构和动力学性质的影响进行了讨论，显示如何循环影响的频谱间隙的大小和同步的倾向。对于特定长度的环，得到了谱的解析公式。
英文摘要：
We derive exact equations that determine the spectra of undirected and directed sparsely connected regular graphs containing loops of arbitrary length. The implications of our results to the structural and dynamical properties of networks are discussed by showing how loops influence the size of the spectral gap and the propensity for synchronization. Analytical formulas for the spectrum are obtained for specific length of the loops.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
