  We investigate the speed of sound and polytropic index of quantum
chromodynamics (QCD) matter in the full phase diagram based on a 3-flavor
Polyakov-looped Nambu-Jona-Lasinio (pNJL) model. The speed of sound and
polytropic index in isothermal and adiabatic cases all have a dip structure at
the low chemical potential side of the chiral phase transition boundary, and
these quantities reach their global minimum values at the critical endpoint
(CEP) but are not completely zero, where the values in adiabatic are lightly
greater than those in isothermal. Different from the speed of sound, the
polytropic index also exists a peak around the chiral phase transition
boundary. Along the hypothetical chemical freeze-out lines, the speed of sound
rapidly decreases near the CEP, followed by a small spinodal behavior, while
the polytropic index, especially in isothermal, exhibits a more pronounced and
nearly closed to zero dip structure as it approaches the CEP.
