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Centaurus A (Cen~A) is the nearest active radio galaxy, which has kiloparsec(kpc) scale jets and {giant lobes detected by various instruments in radio andX-ray frequency ranges}. The $Fermi$--Large Area Telescope and High EnergyStereoscopic System (HESS) confirmed, that Cen~A is a very high-energy (VHE; $>0.1$~TeV) $\gamma$-ray emitter with a known spectral {softening} in the energyrange from a few GeV to TeV. In this work, we consider a synchrotronself-Compton model in the nucleus for the broad band spectrum {below the breakenergy} and an external Compton model in kpc-scale jets for the $\gamma$-rayexcess. Our results show that the observed $\gamma$-ray excess can be suitablydescribed by the inverse Compton scattering of the starlight photons in thekpc-scale jets, which is consistent with the recent tentative report by theHESS on the spatial extension of the TeV emission along the jets. Consideringthe spectral fitting results, the excess can only be seen in Cen~A, which isprobably due to two factors: (1) the host galaxy is approximately 50 times moreluminous than other typical radio galaxies and (2) the core $\gamma$-rayspectrum quickly decays above a few MeV due to the low maximum electron Lorentzfactor of $\gamma_{\rm c}=2.8 \times 10^3$ resulting from the large magneticfield of 3.8~G in the core. By the comparison with other $\gamma$-ray detectedradio galaxies, we found that the magnetic field strength of relativistic jetsscales with the distance from the central black holes $d$ with $B (d) \proptod^{-0.88 \pm 0.14}$.

Inverse Compton scattering of starlight in the kiloparsec-scale jet in Centaurus A: The origin of excess TeV $\gamma$-ray emission