  Designing new 2D systems with tunable properties is an important subject for
science and technology. Starting from graphene, we developed an algorithm to
systematically generate 2D carbon crystals belonging to the family of
graphdiynes (GDYs) and having different structures and sp/sp2 carbon ratio. We
analyze how structural and topological effects can tune the relative stability
and the electronic behavior, to propose a rationale for the development of new
systems with tailored properties. A total of 26 structures have been generated,
including the already known polymorphs such as {\alpha}-, \b{eta}- and
{\gamma}-GDY. Periodic density functional theory calculations have been
employed to optimize the 2D crystal structures and to compute the total energy,
the band structure, and the density of states. Relative energies with respect
to graphene have been found to increase when the values of carbon sp/sp2 ratio
increase, following however different trends based on the peculiar topologies
present in the crystals. These topologies also influence the band structure
giving rise to semiconductors with a finite bandgap, zero-gap semiconductors
displaying Dirac cones, or metallic systems. The different trends allow
identifying some topological effects as possible guidelines in the design of
new 2D carbon materials beyond graphene.
