By Jean Pierre Serre
Jean-Pierre Serre is Professor on the Collège de France. He has written a couple of books, together with "Algebraic teams and sophistication Fields", "Local Fields", "Complex Semisimple Lie Algebras", "Linear Representations of Finite Groups", accumulated Papers (3 volumes), and "Trees" released by means of Springer-Verlag.
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Example text
Those vertices that can be reached by arcs from a vertex a, and the edges incident to them, we call the connected component of a. In this manner, the whole graph falls into connected components with no edges or arcs connecting the separate components. 3, we have illustrated a graph with 4 connected components, one of them an isolated vertex. From the map point of view, we may consider it the road graph of islands, each island having a connected road system. For many considerations in graph theory, we can suppose that the graph is connected, since we can examine separate1y the properties of each connected component.
The result is not simple to prove. It is simple enough for n = 1: if an applicant is qualified for a job, he can be put into it. If n = 2, the diversity condition guarantees that there are at least two jobs for which the two applicants are qualified, and that, moreover, the carpenter-plumber catastrophe does not occur-that is, that each applicant is qualified for at least one job. Once again the applicants can be placed. These remarks show that the theorem is true for n = 1 and for n = 2. It is reasonable to try to prove the theorem in general by the principle of mathematical induction: we shall assume that the statement is true when there are n - 1 applicants or fewer, and we shall deduce that it is true when there are n applicants.
It) the vertex d? (iii) the vertex e? 5 Bracing Frameworks We now use the properties of trees to solve a problem in structural engineering. Many buildings are supported by rectangular steel frameworks, and it is important that such frameworks should remain rigid under heavy loads. 13. ) Is 3 the minimum number of such braces that we must add so as to make this framework rigid? 14. Framework (i) is rigid, but is over-braced since several of the diagonal braces can be removed TREES 47 EE ill EE8 distort add braces non-rigid rigid Figure 313 without affecting the rigidity of the framework.