  We study dynamic interplay between time-delay and velocity alignment in the
ensemble of Cucker-Smale (C-S) particles(or agents) on time-varying networks
which are modeled by digraphs containing spanning trees. Time-delayed dynamical
systems often appear in mathematical models from biology and control theory,
and they have been extensively investigated in literature. In this paper, we
provide sufficient frameworks for the mono-cluster flocking to the continuous
and discrete C-S models, which are formulated in terms of system parameters and
initial data. In our proposed frameworks, we show that the continuous and
discrete C-S models exhibit exponential flocking estimates. For the explicit
C-S communication weights which decay algebraically, our results exhibit
threshold phenomena depending on the decay rate and depth of digraph. We also
provide several numerical examples and compare them with our analytical
results.
