  We present emergent dynamics of continuous and discrete thermomechanical
Cucker-Smale(TCS) models equipped with temperature as an extra observable on
general digraph. In previous literature, the emergent behaviors of the TCS
models were mainly studied on a complete graph, or symmetric connected graphs.
Under this symmetric setting, the total momentum is a conserved quantity. This
determines the asymptotic velocity and temperature a priori using the initial
data only. Moreover, this conservation law plays a crucial role in the flocking
analysis based on the elementary $\ell_2$ energy estimates. In this paper, we
consider a more general connection topology which is registered by a general
digraph, and the weights between particles are given to be inversely
proportional to the metric distance between them. Due to this possible symmetry
breaking in communication, the total momentum is not a conserved quantity, and
this lack of conservation law makes the asymptotic velocity and temperature
depend on the whole history of solutions. To circumvent this lack of
conservation laws, we instead employ some tools from matrix theory on the
scrambling matrices and some detailed analysis on the state-transition
matrices. We present two sufficient frameworks for the emergence of
mono-cluster flockings on a digraph for the continuous and discrete models. Our
sufficient frameworks are given in terms of system parameters and initial data.
