Isospectral graphs with identical nodal counts
  According to a recent conjecture, isospectral objects have different nodal count sequences. We study generalized Laplacians on discrete graphs, and use them to construct the first non-trivial counter-examples to this conjecture. In addition, these examples demonstrate a surprising connection between isospectral discrete and quantum graphs. 
Idan Oren and Ram Band
math-ph math.MP