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Co-based shandite Co$_3$Sn$_2$S$_2$ is a representative example of magneticWeyl semimetals showing rich transport phenomena. We thoroughly investigatemagnetic and transport properties of hole-doped shanditesCo$_3$In$_x$Sn$_{2-x}$S$_2$ by first-principles calculations. The calculationsreproduce nonlinear reduction of anomalous Hall conductivity with doping In forCo$_3$Sn$_2$S$_2$, as reported in experiments, against the linearly decreasedferromagnetic moment within virtual crystal approximation. We show that adrastic change in the band parity character of Fermi surfaces, attributed tothe nodal rings lifted energetically with In-doping, leads to strongenhancement of anomalous Nernst conductivity with reversing its sign inCo$_3$In$_x$Sn$_{2-x}$S$_2$.

First-principles investigation of magnetic and transport properties in hole-doped shandite compounds Co$_3$In$_x$Sn$_{2-x}$S$_2$