A novel slow-wave structure, i.e., an open-style dielectric-lined azimuthally periodic circular waveguide (open-style DLAP-CW) which can be applied to millimeter wave traveling-wave tube, is proposed. The hot dispersion characteristics are derived by the self-consistent relativistic field theory. And the electron beam interaction in the novel slow-wave structure (SWS) is analyzed in a linear frame. The linear gain characteristics of the DLAP-CW is studied analytically for dimensions of the improved SWS and the parameters of the electron beam. The results illustrate that selecting the appropriate dimensions of the metal rods can improve the small-signal gain. Finally, a comparison of the small-signal gain of this structure with a close-style DLAP-CW is made, and the results validate that the novel SWS has an advantage over the close-style DLAP-CW in gain with little influence on the bandwidth, which can potentially improve electron efficiency in the beam wave interaction. The research in this paper will also be a foundation of the theory for open-style dielectric-lined azimuthally periodic circular waveguide traveling-wave tube.