The aluminum substructure of the AeM(2)Al(9) (Ae = Ba and M = Fe, Co, Ni; Ae = Sr and M = Co, Ni; Ae = Ca and M = Co) and CaNiAl9 compounds is a beautiful three-dimensional network of vertex-sharing aluminum octahedra. Bonding in this network is analyzed at the extended Huckel level by studying the effect of vertex sharing between isolated aluminum octahedral clusters in several model systems : a linear dimer of two clusters, a linear one-dimensional chain of clusters, a two-dimensional square sheet, and a three-dimensional cubic network of clusters. We find that the number of skeletal electrons per aluminum cluster optimal for Al-Al bonding is reduced from 14 for an isolated cluster to 12 for the cluster dimer and cluster chain, 10 for the two-dimensional cluster sheet, and 8 for the cubic network in which all cluster vertices are shared. Two effects are responsible for the reduction in the optimum electron count : First, vertex-sharing reduces the number of skeletal bonding orbitals per cluster (through restrictions due to translational symmetry in extended structures). Second, the levels just above the skeletal bonding states become Al-Al antibonding due to next-nearest-neighbor intercluster interactions. According to our calculations, Al-Al bonding in the aluminum network of the BaFe2Al9-type compounds is maximized for approximately 9 skeletal electrons per aluminum octahedral cluster, which is qualitatively consistent with the results obtained for the model systems and our assignment of formal charges. Connections are made to a number of related structures containing networks of main group octahedra.