  We introduce a simplified model for a three-terminal quantum thermocouple
consisting of two strongly-coupled quantum dots. To elucidate spin-dependent
Seebeck and Peltier effects, we employ a microscopic Hamiltonian and map the
Lindblad master equation onto a quantum transition network, capturing the key
working principles for both reciprocal effects. Our analysis reveals quantum
thermodynamic networks encompassing both Coulomb interaction and spin-flipping
processes, lead to the emergence of spin-thermolectric effects. Using algebraic
graph theory, we recover the phenomenological law of irreversible
thermodynamics from the stochastic version of the entropy production rate
expressed in terms of cycle flux and cycle forces. Remarkably, Onsager
reciprocity and Kelvin relation for transport coefficients find their premises
in the properties of cycle flux trajectories within the quantum transition
network. This underscores the universal generality of thermodynamic principles
across classical and quantum realms, despite their fundamentally different
basis from classical laws of irreversible thermodynamics relying on local
equilibrium assumptions.
