  We investigate the cosmological production of gravitational waves in a
nonsingular flat cosmology powered by a "running vacuum" energy density
described by $\rho_{\Lambda}\equiv\rho_{\Lambda}(H)$, a phenomenological
expression potentially linked with the renormalization group approach in
quantum field theory in curved spacetimes. The model can be interpreted as a
particular case of the class recently discussed by Perico et al. (Phys. Rev. D
{\bf 88}, 063531, 2013) which is termed complete in the sense that the cosmic
evolution occurs between two extreme de Sitter stages (early and late time de
Sitter phases). {The gravitational wave equation is derived and its
time-dependent part numerically integrated since the primordial de Sitter
stage. The generated spectrum of gravitons is also compared with the standard
calculations where an abrupt transition, from the early de Sitter to the
radiation phase, is usually assumed.} It is found that the stochastic
background of gravitons is very similar to the one predicted by the cosmic
concordance model plus inflation except at higher frequencies ($\nu \gtrsim
100$ kHz). This remarkable signature of a "running vacuum" cosmology combined
with the proposed high frequency gravitational wave detectors and measurements
of the CMB polarization (B-modes) may provide a new window to confront more
conventional models of inflation.
