单模网络谱的一个下界
A lower bound on the spectrum of unimodular networks
AI总结：
本文根据平均度证明了单模网络邻接算子与Markov算子谱半径的下界，进而推广了正则图的Alon-Boppana定理，并建立了单模树谱半径与体积增长率的下界。
摘要：
单模网络是有限图在随机意义上的一种推广。我们根据单模网络的平均度，证明了其邻接算子和Markov算子谱半径的一个下界。这允许我们针对大型、连通且度有界的图，证明其绝对值最大的特征值满足Alon-Boppana型界，从而推广了关于正则图的Alon-Boppana定理。关键步骤是根据平均度，建立单模树谱半径的一个下界。类似地，我们根据平均度，提供了单模树体积增长率的一个下界。
英文摘要：
Unimodular networks are a generalization of finite graphs in a stochastic sense. We prove a lower bound to the spectral radius of the adjacency operator and of the Markov operator of an unimodular network in terms of its average degree. This allows to prove an Alon-Boppana type bound for the largest eigenvalues in absolute value of large, connected, bounded degree graphs, which generalizes the Alon-Boppana theorem for regular graphs. A key step is establishing a lower bound to the spectral radius of a unimodular tree in terms of its average degree. Similarly, we provide a lower bound on the volume growth rate of an unimodular tree in terms of its average degree.
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