Graph Theory is used to explain famous problems like Traveling Salesman's Problem and Chinese Postman Problems. Weighted Graphs are widely used in solving distance related problems as well. What are the significant uses of graph theory and graph labeling techniques in Transportation and related areas?
This question is really too wide to answer succinctly. There is a vast amount of problems and applications associated with graphs - hence there are tons of mathematical models, analyses, and algorithms, too. In transportation engineering we may be interested in knowing, for example, the robustness and vulnerability of a graph ("if one link brakes, can you still go from A to B?") - and the associated analysis of a robust network construction. Graph theory and methods are used all the time in the construction and assessment of new road constructions - how will an interference in a traffic network (including less capacity, as discussed above, as well as added capacity) influence traffic and traveller behavior? Equilibrium network design is one such discipline, in which one studies the best interference/design of a traffic network in order to obtain given goals. "Design" can here mean both a physical re-design of the network and the design of pricing mechanisms in order to influence how - and when - people travel. The field is vast and still need to be further explored, I would say.
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