  By means of a slight modification of the notion of GM-complexity, the present
paper performs a graph-theoretical approach to the computation of (Matveev's)
complexity for closed orientable 3-manifolds. In particular, the existing
crystallization catalogue C^{28}, due to Lins, is used to obtain upper bounds
for the complexity of closed orientable 3-manifolds triangulated by at most 28
tetrahedra. The experimental results actually coincide with the exact values of
complexity, for all but three elements. Moreover, in the case of at most 26
tetrahedra, the exact value of the complexity is shown to be always directly
computable via crystallization theory.

Geometric Topology