Lower bound on pebbling numbers
A full proof (based on superconcentrators) can be found in chapter 24 "The pebble game" of the book


  Uwe Schöning and Randall Pruim:
  Gems of Theoretical Computer Science
  Springer, 1998  
  
  ISBN 978-3-642-64352-1
  https://link.springer.com/book/10.1007%2F978-3-642-60322-8