  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.
