Course Content (Syllabus)
1. COUNTING TECHNIQUES: Product Principle, The Sum Principle, Permutations - Lists - Combinations, with and without repetition, Binomial Coefficients, Pigeonhole or Dirichlet Principle, Inclusion-Exclusion Principle, derangements, Reflection Principle, Routs in Grids, The lexicographic permutation order.
2. SPECIAL TOPICS IN COUNTING: The Pascal’s Triangle and the Fibonacci Numbers, Diophantine equations and Partitions, Distribution Problems (Beads in Cells, Stirling's, Bell, Catalan Numbers), Generator Functions.
3. GRAPHS: Basic Concepts (order, size, connectivity, direction, neighbors, walk, path, trail, circle, complement, bipartite graphs, operations, degree, geodesics, distance, diameter, radius), Properties, Matrices, Isomorphism, line graph), Subgraphs, Paths, Trees, Factors, Bridges, Theorems of Kirchhoff, Dirac, Menger. Special Graphs (plane graphs, Euler, Hamilton, n-cubes, Gray Codes, Ramsey Numbers ), Colors (basic theorems, color polynomials, algorithms).
4. INTRODUCTION TO Random Graphs: Erdös-Rényi Networks (Degree Distribution, Average degree, Giant Component, Average Distance, Clustering Coefficient, Transitivity), Introduction to Small World and Scale Free Networks, Introduction to Real Networks, , Centrality: degree, eigenvalue, closeness, betweenness), Examples Using the R Language (Collaboration Networks, Social, Financial, Online, etc.)