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Beginning Functional Analysis, Karen Saxe. Calculus of Several Variables, Serge Lang. Combinatorial Optimization for Undergraduates, L. Combinatorics and Graph Theory, John M. Harris Jeffry L. Hirst Michael J. Complex Analysis, Theodore W. Conics and Cubics, Robert Bix. Differential Equations, Clay C. Discrete Mathematics, L.

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Primer of Modern Analysis, Kennan T. Silverman John Tate. Second Year Calculus, David M. Short Calculus, Serge Lang. This completes the induction.


The next theorem extends the preceding result to forests. The proof is similar and appears as Exercise 4.

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If F is a forest of order n containing k connected components, then F contains n — k edges. The next two theorems give alternative methods for defining trees. Two other methods are given in Exercises 5 and 6. A graph of order n is a tree if and only if it is connected and contains n — 1 edges.

The forward direction of this theorem is immediate from the definition of trees and Theorem 1. For the reverse direction, suppose a graph G of order n is connected and contains n — 1 edges. We need to show that G is acyclic. If G did have a cycle, we could remove an edge from the cycle and the resulting graph would still be connected.

In fact, we could keep removing edges one at a time from existing cycles, each time maintaining connectivity. The resulting graph would be connected and acyclic and would thus be a tree. But this tree would have fewer than n — 1 edges, and this is impossible by Theorem 1.