One of the unexpecting emerging interactions between seemingly distant areas in mathematics is between graph theory and analysis.  One such link is the theory of continuous limits of discrete structures.  This theory has applications in computer science, probability theory, the theory of quasirandomness, number theory, statistical physics, and elsewhere.  Further examples of interactions include the theory of measurable graphs, connections with ergodic theory, function norms related to graphs, analytic methods in extremal graph theory, and differential equations on graphs.