图的泛覆盖的非回溯谱
The Non-Backtracking Spectrum of the Universal Cover of a Graph
摘要：
图$H$上的非回溯行走是$H$的有向边的有向路径，使得没有边是其前一条边的逆。给定长度的非回溯遍历可以使用非回溯邻接矩阵$B$来计数，该矩阵由$H$的有向边索引并且与Ihara的Zeta函数相关。我们展示了如何确定$B$的频谱的情况下，$H$是一棵树覆盖有限图。我们证明了当$H$不正则时，这个谱在复平面上可以有正测度，而不像正则的情形.我们发现，在$B$的频谱之外，相应的格林函数具有周期性的衰减比。“这样的‘比率体系’的存在是可以被有效遏制的，相当于游离于光谱之外。我们还证明了覆盖有限图的树上的非回溯行走算子的谱半径恰好是$\sqrt\gr$，其中$\gr$是树的增长率.这进一步激发了Stark和Terras \cite{ST}提出的图论黎曼假设的定义。最后，我们给出了实验证据，对于一个固定的，有限的图，$H$，一个大的程度的随机提升有非回溯新的频谱附近的$H$的通用覆盖。这表明一个新的推广阿隆的第二特征值猜想。
英文摘要：
A non-backtracking walk on a graph, $H$, is a directed path of directed edges of $H$ such that no edge is the inverse of its preceding edge. Non-backtracking walks of a given length can be counted using the non-backtracking adjacency matrix, $B$, indexed by $H$'s directed edges and related to Ihara's Zeta function. We show how to determine $B$'s spectrum in the case where $H$ is a tree covering a finite graph. We show that when $H$ is not regular, this spectrum can have positive measure in the complex plane, unlike the regular case. We show that outside of $B$'s spectrum, the corresponding Green function has ``periodic decay ratios.'' The existence of such a ``ratio system'' can be effectively checked, and is equivalent to being outside the spectrum. We also prove that the spectral radius of the non-backtracking walk operator on the tree covering a finite graph is exactly $\sqrt\gr$, where $\gr$ is the growth rate of the tree. This further motivates the definition of the graph theoretical Riemann hypothesis proposed by Stark and Terras \cite{ST}. Finally, we give experimental evidence that for a fixed, finite graph, $H$, a random lift of large degree has non-backtracking new spectrum near that of $H$'s universal cover. This suggests a new generalization of Alon's second eigenvalue conjecture.
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