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.

翻译：单模网络是有限图在随机意义下的推广。我们证明了单模网络邻接算子和马尔可夫算子的谱半径关于其平均度的下界。由此可证明具有大尺度、连通且有界度的图中绝对值最大特征值的Alon-Boppana型下界，这推广了正则图的Alon-Boppana定理。关键步骤是建立单模树谱半径关于其平均度的下界。类似地，我们给出了单模树体积增长率关于其平均度的下界。