Due to the diversified atomic structures and electronic properties, two-dimensional monolayer nanocarbon materials (graphyne or graphdiyne) composed of sp and sp2 hybridization C atoms have received the widespread attention in recent years. The fundamental questions include how the sp orbital hybridization affects the electronic structure of graphyne. In order to investigate the structure dependent electronic structures of graphyne, the energetic stabilities and electronic structures of -graphyne and its derivatives (-N) with N carbon atoms on each edge of the hexagons are investigated by density functional theory (DFT) calculations in this work. In our DFT calculations we adopt generalized gradient approximation of Perdew, Burke, and Ernzerhof (GGA-PBE) using the CASTEP module implemented in Materials Studio. The studied -Ns consist of hexagon carbon rings connected by vertexes whose edges have various numbers of carbon atoms N= 1-10. The structure and energy analyses show that -Ns with even-numbered carbon chains have alternating single and triple C-C bonds, energetically more stable than those with odd-numbered carbon chains possessing continuous C-C double bonds. The calculated electronic structures indicate that -Ns can be either metallic (odd N) or semiconductive (even N), depending on the parity of number of hexagon edge atoms regardless of the edge length due to Jahn-Teller distortion effect. Some semiconducting -graphyne derivatives (-N, N= 2, 6, 10) are found to possess Dirac cones (DC) with small direct band gaps 10 meV and large electron velocities 0.255106-0.414106 m/s, ～30%-50% of that of graphene. We find that Dirac cones also appear in -3 and -4 when we shorten the double bonds and elongate the triple bonds in -3 and -4 respectively. These results show that the bond length change will affect the characteristics of band structure and suggests that the band structure characteristics may be influenced by Peierls distortion in a two-dimensional system. Our DFT studies indicate that introducing sp carbon atoms into the hexagon edges of graphene opens the way to switching between metallic and semiconductor/DC electronic structures via tuning the parity of the number of hexagon edge atoms without doping and defects in nanocarbon materials and nanoelectronic devices.