Applying the Briggs-Bers " pole-pinch " criterion to the exact transcendental dispersion relation of a dielectric traveling wave tube (TWT), we find that there is no absolute instability regardless of the beam current. We extend this analysis to the circuit band edges of a linear beam TWT by approximating the circuit mode as a hyperbola in the frequency-wave-number (ω-k) plane and consider the weak coupling limit. For an operating mode whose group velocity is in the same direction as the beam mode, we find that the lower band edge is not subjected to absolute instability. At the upper band edge, we find a threshold beam current beyond which absolute instability is excited. The nonexistence of absolute instability in a linear beam TWT and the existence in a gyrotron TWT, both at the lower band edge, is contrasted. The general study given here is applicable to some contemporary TWTs such as metamaterial-based and advanced Smith-Purcell TWTs.