Is there any physical interpretation of the constant which is seen in the constraint curve of an adiabatic process?

Start your process with the gas having volume and pressure $V_0, p_0$. If the process is adiabatic and reversible then $pV^gamma = p_0V_0^gamma$, in other words $C=p_0V_0^gamma$.

$C$ is a constant. It simply means that the relationship on the left between pressure, volume and the ratio of specific heats of constant pressure and volume yields the same number at any equilibrium state during the process. The physical interpretation in the case of this process is that it is a reversible adiabatic (isentropic, or constant entropy) process for an ideal gas.

The derivation of this equation is based on no change in entropy, coupled with the ideal gas equation, definition of enthalpy and internal energy, and assumption of constant specific heats. For a derivation see:

http://www.mhtl.uwaterloo.ca/courses/me354/lectures/pdffiles/ch2.pdf

As @Knzhou pointed out although the process is constant entropy, $C$ is not the value of the entropy.

A similar situation exists for a reversible isothermal (constant temperature) process. The equation is

$$PV=C$$

Where again $C$ is a constant (not the same constant as the constant entropy process) but it is not the value of the constant temperature. Here

$$C=nRT$$

Both the isentropic and reversible isothermal process are specific cases of the more general reversible polytropic process for an ideal gas, where

$$PV^{n}=C$$

For the isentropic process, $n=C_{p}/C_{v}$. For the isothermal process, $n=1$. For a constant pressure (isobaric) process $n=0$.

Hope this helps.