On the ground state of quantum graphs with attractive $\delta$-coupling
  We study relations between the ground-state energy of a quantum graph Hamiltonian with attractive $\delta$ coupling at the vertices and the graph geometry. We derive a necessary and sufficient condition under which the energy increases with the increase of graph edge lengths. We show that this is always the case if the graph has no branchings while both change signs are possible for graphs with a more complicated topology. 
Pavel Exner and Michal Jex
math-ph math.MP math.SP quant-ph