Graph theory, combinatorics, and algorithmic design. She is an authority on spectral-graph theory, extremal graphs, graph labeling, graph decompositions, random graphs, graph algorithms (all useful for visualizing and modeling networks), and parallel structures. One focus is whether common mathematical models can describe multiple types of networks, not only computer networks such as the Internet but also social networks connecting people, and biological models of gene and protein interactions. A related matter is how accurately distance values, such as the average and maximum "hops" a data packet travels to destination, can be predicted using relatively few parameters. Graham has written or co-written numerous papers in areas ranging from pure mathematics ( e.g., differential geometry, number theory) to the applied (e.g., optimization, computational geometry, telecommunications and Internet computing). She has developed courses to bridge the gap between classroom mathematics, and math practiced in the real world.
