A random hierarchical lattice: the series-parallel graph and its properties

Journal article

We consider a sequence of random graphs constructed by a hierarchical procedure. The construction replaces existing edges by pairs of edges in series or parallel with probability $p$ and $1-p$ respectively. We investigate the effective resistance across the graphs, first-passage percolation on the graphs and the Cheeger constants of the graphs as the number of edges tends to infinity. In each case we find a phase transition at $p=1/2$.

Authors: Hambly, B; Jordan, J

• Publication date: 2003-01-01