A populous group of ternary trielide rich (Al, Ga, In) intermetallics forming the NaZn13 structure type has been synthesized from stoichiometric combinations of the elements in an are melter. These ternary compounds have the general formula AM(x)T(13-x), where A = Ba, Sr, La, Eu, M = Cu and Ag, and T = Al, Ga, and In, with 5 less than or equal to x less than or equal to 6.5, and have been structurally characterized by both powder and single-crystal X-ray diffraction. Furthermore, magnetic susceptibility, electrical resistivity, XPS, and EDS measurements are reported for some of the samples. Single-crystal X-ray diffraction experiments on BaCu5Al8 (BaCu5.10(7)Al-7.90(7), cubic, a = 12.205(4) Angstrom, Z = 8) and EuCu6.5Al6.5 (EuCu6.41(5)Al-6.59(5), cubic, a = 11.928(1) Angstrom, Z = 8) indicate that the quasi-infinite three-dimensional [CuxAl13-x] framework involves mostly Cu atoms centering icosahedra, with its vertexes randomly occupied by the remaining Cu and Al atoms. On the other hand, when M = Ag, Al shows a greater tendency to occupy the center of the icosahedra. A systematic study of the compositional variation in BaCuxAl13-x demonstrates that the NaZn13 type phase exists within a narrow range of x between five and six. To examine the role of the cation A in stabilizing this structure, quaternary phases, e.g., BaSrAg12Al14, (BaSrAg12.0(1)Al-14.0(1) cubic, a = 12.689(1) Angstrom, Z = 4) and SrCeCu12Al14 (SrCeCu11.74(2)Al-14.26(2), cubic, a = 11.938(1) Angstrom, Z = 4), were prepared and characterized. Extended Huckel calculations on these ternary aluminides demonstrate how the tuning of the system's stoichiometry maximizes the bonding within the atom-centered icosahedral framework. These calculations also address the substitution pattern of M and T within the [MxT13-x] network. Tight-binding LMTO calculations have also been applied to examine the charge-density and electron localization functions (ELF) in this structure for different electron counts in order to address the nature of chemical bonding in these phases. One important conclusion from the theoretical results is that the NaZn13 type phases show optimal stability for 40-42 valence electrons for the [MxT13-x] framework.