Evidence of the FIP effect in the coronae of late-type giants

$\beta$ Cet, 31 Com and $\mu$ Vel represent the main stages through which late-type giants evolve during their lifetime (the Hertzsprung gap (31 Com), the rapid braking zone ($\mu$ Vel) and the core helium burning ``clump'' phase ($\beta$ Cet)). An analysis of their high resolution {\it Chandra} X-ray spectra reveals similar coronal characteristics in terms of both temperature structure and element abundances for the more evolved stars ($\mu$ Vel and $\beta$ Cet) with slight differences for the `younger' giant (31 Com). The coronal temperature structure of 31 Com is significantly hotter showing a clear peak while $\beta$ Cet and $\mu$ Vel show a plateau. $\beta$ Cet and $\mu$ Vel show evidence for a FIP effect in which coronae are depleted in high FIP elements relative to their photospheres by a factor of $\sim 2$. In contrast, 31 Com is characterized by a lack of FIP effect. In other words, neither depletion nor enhancement relative to stellar photospheric values is found. We conclude that the structural changes during the evolution of late-type giants could be responsible for the observed differences in coronal abundances and temperature structure. In particular, the size of the convection zone coupled with the rotation rate seem obvious choices for playing a key role in determining coronal characteristics.