On ergodic diffusions on continuous graphs whose centered resolvent admits a trace

We consider ergodic and reversible diffusions on continuous and connected graphs GG with a finite number of bifurcation vertices and some rays going to infinity. A necessary and sufficient condition is presented for the spectrum of the associated generator L to be without continuous part and for the sum of the inverses of its eigenvalues (except 0) to be finite. This criterion is easily computable in terms of the coefficients of L   and does not depend on the transition kernels at the vertices. Its motivation is that it is conjectured to be also a necessary and sufficient condition for the diffusion to admit strong stationary times whatever its initial distribution (this is known to be true if GG is the real line). The above criterion for the centered resolvent to be of trace class is next extended to Markov processes on denumerable connected graphs with only a finite number of vertices of degree larger than or equal to 3.