Understanding the magnetism and its possible correlations to the topological properties has emerged as a forefront and difficult topic in studying magnetic Weyl semimetals. Co$_{3}$Sn$_{2}$S$_{2}$ is a newly discovered magnetic Weyl semimetal with a kagome lattice of cobalt ions and has triggered intense interest for rich fantastic phenomena. Here, we report the magnetic exchange couplings of Co$_{3}$Sn$_{2}$S$_{2}$ using inelastic neutron scattering and two density functional theory (DFT) based methods: constrained magnetism and multiple-scattering Greens function methods. Co$_{3}$Sn$_{2}$S$_{2}$ exhibits highly anisotropic magnon dispersions and linewidths below $T_{C}$, and paramagnetic excitations above $T_{C}$. The spin-wave spectra in the ferromagnetic ground state is well described by the dominant third-neighbor across-hexagon $J_{d}$ model. Our density functional theory calculations reveal that both the symmetry-allowed 120$^circ$ antiferromagnetic orders support Weyl points in the intermediate temperature region, with distinct numbers and the locations of Weyl points. Our study highlights the important role Co$_{3}$Sn$_{2}$S$_{2}$ can play in advancing our understanding of kagome physics and exploring the interplay between magnetism and band topology.
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