The fragility index (m) of a series of Ge-As-S glasses covering a wide range of mean coordination (< r >) and stoichiometry is measured by differential scanning calorimetry. The evolution of the fragility index appears to be better predicted by the stoichiometry than the mean coordination, and m shows a well-defined dependence on the departure from stoichiometry quantified as the at. % excess or deficiency in sulfur. The effect of stoichiometry on the fragility of Ge-As-S glasses is very similar to that observed in the Ge-As-Se system. A systematic Raman spectroscopy investigation also indicates the presence of large fractions of molecular species such as S-8, As4S4, and As4S3 in the structure of some glasses. The presence of molecular species is shown to increase the fragility and decrease the glass transition temperature (T-g). These results emphasize that short-range order rather than long-range characteristics such as structural rigidity appears to control the fragility of chalcogenide glasses.