Geometry and spectrum of rapidly branching graphs
  We study graphs whose vertex degree tends and which are, therefore, called rapidly branching. We prove spectral estimates, discreteness of spectrum, first order eigenvalue and Weyl asymptotics solely in terms of the vertex degree growth. The underlying techniques are estimates on the isoperimetric constant. Furthermore, we give lower volume growth bounds and we provide a new criterion for stochastic incompleteness. 
Matthias Keller, Felix Pogorzelski, Florentin M\"unch
math.SP math.CO