We calculate the photonic band gap of triply periodic bicontinuous cubic structures and of tubular structures constructed from the skeletal graphs of triply periodic minimal surfaces. The effect of the symmetry and topology of the periodic dielectric structures on the existence and the characteristics of the gaps is discussed. We find that the C(I-2-Y-**) structure with Ia (3) over bard symmetry, a symmetry which is often seen in experimentally realized bicontinuous structures, has a photonic band gap with interesting characteristics. For a dielectric contrast of 11.9 the largest gap is approximately 20% for a volume fraction of the high dielectric material of 25%. The midgap frequency is a factor of 1.5 higher than the one for the (tubular) D and G structures. For a volume fraction of 25% the smallest dielectric contrast required to open a gap for the C(I-2-Y-**) structure is 4.5. A gap of width larger than 10% is obtained with dielectric contrasts of 7 and higher.
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