Almost forty years ago I was persuaded that combinatorial isoperimetric problems were worthy of systematic investigation. The edge- and vertexisoperimetric problems were clearly fundamental aspects of graph theory. They had already been applied to the wirelength and bandwidth problems on d-cubes and other graphs which had engineering implications. As analogs of the classical isoperimetric problem of Greek geometry they seemed certain to lead to further useful results. Over the years this analogy, with the pressure of prospective applications, has produced profound solution methods; spectral, global and variational.

It has been very difficult to bring closure to the writing of this monograph since every time I go over the material, new insights appear and demand to be included. Also, tempting new problems keep arising in science, engineering and mathematics itself. For instance, Lubotzky's monograph [75] has a whole chapter of unsolved problems. It seems certain that the subject will continue to progress for the foreseeable future, but life is short and we cannot wait until every significant question has been answered. Last week, in a conversation with T. H. Payne, colleague, collaborator and for many years a most reliable source of information about trends in computer science, I mentioned recent work on the profile scheduling problem (see Chapter 8). “Oh, yes,” he said with enthusiasm, “that has been applied to optimizing straight-line programs! A ‘live variable’ must be stored in a register, so the profile equals total storage time. But the latest thing is to minimize register width, the maximum number of registers required by a program.”