Laplacians on Metric Graphs: Eigenvalues, Resolvents and Semigroups
  The main objective of the present work is to study the negative spectrum of (differential) Laplace operators on metric graphs as well as their resolvents and associated heat semigroups. We prove an upper bound on the number of negative eigenvalues and a lower bound on the spectrum of Laplace operators. Also we provide a sufficient condition for the associated heat semigroup to be positivity preserving. 
Vadim Kostrykin and Robert Schrader