Discrete thermodynamic Cucker-Smale model with time-delay on a general digraph

We study the discrete-time thermodynamic Cucker-Smale (TCS) model with a communication time-delay on a general digraph containing a spanning tree. In the TCS model, the mutual interactions not only are "mechanical" but also are affected by the "temperature effect." It always takes time for information to be received between agents due to finite propagation speed effects. Transmission delays are inevitable and should be incorporated into flocking modeling. In this paper, we provide sufficient frameworks for flocking to the discrete TCS model, which are formulated in terms of initial configuration, network topology, and system parameters. In our proposed frameworks, we show that the TCS model exhibits exponential flocking convergence.