The subgaussian constant of a graph arises naturally in bounding the moment generating function of Lipschitz functions on the graph, with a given probability measure on the set of vertices. The closely related spread constant of a graph measures the maximal variance over all Lipschitz functions on the graph. As such they are both useful (as demonstrated in the works of Bobkov-Houdré-Tetali and Alon-Boppana-Spencer) for describing the concentration of measure phenomenon in product graphs. An equivalent formulation of the subgaussian constant using a transportation inequality, introduced by Bobkov-Götze, is investigated here in depth, leading to a new way of bounding the subgaussian constant. A tight concentration result for the discrete torus is given as a concrete application. An infinite family of graphs is also provided here to demonstrate that typically the spread and the subgaussian constants differ by an order of magnitude.