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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.

Books in the Mathematical Sciences

Lovasz J. Pelikan K. Discrete Probability, Hugh Gordon. Elementary Topics in Differential Geometry, J. Elements of Algebra, John Stillwell. Elements of Number Theory, John Stillwell. Factorization and Primality Testing, David M. Functions of Several Variables, Wendell Fleming. General Topology, Jacques Dixmier. Geometric Constructions, George E. Geometry, Richard S. Millman George D. Geometry: Euclid and Beyond, Robin Hartshorne. Geometry: Plane and Fancy, David A. Glimpses of Algebra and Geometry, Gabor Toth. Groups and Symmetry, M. Inside Calculus, George R. Intermediate Calculus, Murray H.

Intermediate Real Analysis, Emanuel Fischer. LeCuyer Jr. Introduction to Cryptography, Johannes A. Introduction to Linear Algebra, Serge Lang.

Math books and ebooks added July 2018

Introduction to Mathematical Logic, Jerome Malitz. Introduction to the Mathematics of Finance, Steven Roman. Singer J. Lectures on Boolean Algebras, Paul R. Limits, Alan F. Linear Algebra, Charles W.

Linear Algebra, Larry Smith. Linear Algebra, Serge Lang. Linear Algebra, Robert J. Linear Algebra, Klaus Janich. Mathematical Analysis, Andrew Browder. Mathematical Logic, H. Ebbinghaus J. Flum W. Mathematics and Its History, John Stillwell. Methods of Mathematical Economics, Joel Franklin.

Naive Set Theory, Paul R. Notes on Set Theory, Yiannis N. Numbers and Geometry, John Stillwell. Optimization Techniques, L.

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

References

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.