1. Syllabus.

2. An update of the course content covered a session-by-session basis.

3. The course will be partially lectured in English. This is the Schedule for the Lectures in English. Since English is not your native language you are stronglyencouraged to read in advance the corresponding material.

4. Textbook and Reading Materials.

5. Homework Assignments.

6. Test Dates and Solutions.

7. Related Software Links.

8. Related online Journals Links.

9. Links to online Free Courses Related to Graph Theory.

10. Inspirational and Confusing unrelated material intended to push out-of-the-box thinking and free association of ideas.

Here is a list of some online journals, publishing state of the art research in Graph Theory, Combinatorics and related fields.

OPEN ACCESS JOURNALS

You can access the full content of the following journals online, for free.

1. Applicable Analysis and Discrete Mathematics.

2. The Electronic Journal of Combinatorics.

3. Online Journal of Analytic Combinatorics.

4. Séminaire Lotharingien de Combinatoire.

5. International Journal of Mathematical Combinatorics.

6. INTEGERS. Electronic Journal of Combinatorial Number Theory.

7. International Journal on Applications of Graph Theory in Wireless Ad Hoc Networks and Sensor Networks (GRAPH-HOC).

8. International Journal of Mathematics and Soft Computing.

LIMITED ACCESS JOURNALS

You can partially access the content of these journals, through the data basis of the university hosted in SINAB. All the following journals belong to the Springer Journal. You can log in to here.

1. Combinatorica.

2. Graphs and Combinatorics.

3. Annals of Combinatorics.

4. Journal of Algebraic Combinatorics.

5. Algoritmica.

1. August 5th, 2014. Introduction of the course. Evaluation system. Bibliography. Prerequisite and scope lecture on Graph Theory in general and of the course in particular. Homework 0: Installation of MiKTeX, of Kile/TeXMaker, and successfully running the template. Find the direction on this link.

2. August 7th, 2014. (Lecture in English). Homework 1. Read section 1.1, pages 1-4 from Bondy & Murty. Read section 1.1, pages 2-4 from Reinhard Diestel.

3. August 12th, 2014. Main definitions, section 1.1, Vertex degrees section 1.5, paths section 1.6, Bondy & Murty. Minimum, maximum, average degree, page 5, Graph Theory. Reinhard Diestel.

4. August 14th, 2014. (Lecture in English).The triangle free graph (Mantel 1907), page 6 Bollobas. Regular Graphs, pages 16, 17, the construction of Harary graphs, pags 226, 227, Graph Theory and its Applications. Gross & Yellen. First Draft of Homework 1 available.

5. August 19th, 2014. The triangle-free graph and Harary graphs, review and deepening details, see Explanatory Notes. Subgraphs, Spanned (EXHAUSTIVO) and Induced graphs, page 4, second paragraph Graph Theory. Reinhard Diestel. Propositions 1.2.2 and 1.3.1 Graph Theory. Reinhard Diestel.

6. August 21st, 2014. (Lecture in English). Propositions 1.3.1, 1.3.2, 1.3.3 Graph Theory. Reinhard Diestel. Theorem 3, pg 9 Modern Graph Theory. Béla Bollobás. Corollary: The distance in a graph is a metric, see Explanatory Notes.

7. August 22th, 2014. Debate Session 1. Discussion of definition 1, problem 2, problem 4, problem 20. Second Draft Homework 1, available.

8. August 24th, 2014. Debate Session 2. Discussion of problems 4 and 5. The Professor was absent due to exceptional circumstances.

9. August 26th, 2014. Distance in a graph (see Explanatory Notes). Definition of radius and central vertex. Bounding the radius with diameter and vice versa Definition of connected components. Characterization of connected graphs with existence removable vertices (see Explanatory Notes).

10. August 28th, 2014. (Lecture in English). Proposition 1.3.3 , 1.4.1 Graph Theory. Reinhard Diestel. Bounding from below the order of the graph with the girth and the minimum degree. See pgs 9, 10 Graph Theory. Reinhard Diestel. Bipartite graphs, definition and examples See pgs 17, 18 Graph Theory. Reinhard Diestel. Theorem 1.3.19, see Introduction to Graph Theory. Douglas B. West. Homework 1 available.

11. September 1st, 2014. Debate Session 3. Discussion of problems 5, 7 and 14.

12. September 2nd, 2014. Bipartite graphs, theorem 1.2, section 1.7 Graph Theory With Applications. Bondy & Murty. See the claim on odd cycles in the Explanatory notes. Application of connectedness to rigidity, see Graph Theory and its Applications. Gross & Yellen.

13. September 4th, 2014. (Lecture in English). Discussion on the claim for odd cycles and bipartite graphs characterization proof. Trees, theorems 2,1 (two different proofs) and intermediate lemmas, Graph Theory With Applications. Bondy & Murty. Theorem 1.5.1, Graph Theory. Reinhard Diestel. Theorem 5, Theorem 6, Modern Graph Theory. Béla Bollobás.

14. September 8th, 2014. Debate Session. Discussion on the minimum order of a graph, based on its girth and its minimum degree, pages 9, 10, Graph Theory. Reinhard Diestel. Problem 12 of Homework 1, plus discussion of the asymptotic behavior of the bound for large graphs verifying the hypothesis of the problem.

15. September 9th, 2014. Theorem 2.2, corollary 2.2, section 2.1 (two different proofs) Graph Theory With Applications. Bondy & Murty. Corollary 7, page 10, Modern Graph Theory. Béla Bollobás.

16. September 11th, 2014. (Lecture in English). Existence of spanning trees for connected graphs, alternative proof. Corollaries 1.5.2, 1.5.3, 1.5.4. Graph Theory. Reinhard Diestel.

17. September 15th, 2014. Debate Session. The session is suspended as a break after the FIRST TEST. Homework 2, First Draft, available.

18. September 16th, 2014. Class canceled due to lack of quorum. Homework 2, Second Draft, available.

19. September 18th, 2014. (Lecture in English). Rooted trees, down closure, up closure, partial ordering of a rooted tree, page 17, Graph Theory. Reinhard Diestel. Proposition 1.6.1, Graph Theory. Reinhard Diestel. Definitions: parents, children, siblings, ancestors, descendents and m-ary trees. Counting and estimating number of vertices in m-ary trees, pgs 126, 127, Graph Theory and its Applications. Gross & Yellen.

20. September 22nd, 2014. Debate Session. Discussion on dyadic numbers, problems 1 and 2 from Problem Set II.

21. September 23th, 2014. Binary trees, main definitions and recursive property. Theorem 3.2.5, Corollary 3.2.6 Graph Theory and its Applications. Gross & Yellen. Eulerian graphs, definitions, proof of the necessity condition and characterization of semi-Eulerian graphs pages 14, 15, 16 Modern Graph Theory. Béla Bollobás. See also Theorem 6 in the Explanatory Notes. Homework 2, available.

22. September 25th, 2014. Suspended due to Semana Universitaria Break.

23. September 29th, 2014. Debate Session. The Professor will be absent due to mandatory meeting, however, the students are strongly encouraged to attend.

24. September 30th, 2014. Eulerian Graphs characterization. Theorem 1 page 5, Theorem 12 page 16, Modern Graph Theory. Béla Bollobás; Theorem 2.4.2, Introduction to Graph Theory. Douglas B. West.

25. October 2nd, 2014. (Lecture in English) Amend of the proof shown in class, September 23th for semi-Eulerian tours, Theorem 2.4.3, Introduction to Graph Theory. Douglas B. West. Planar Graphs, basic definitions, pages 20, 21; Euler's Formula Theorem 15 page 22, Modern Graph Theory. Béla Bollobás.

26. October 6th, 2014. Debate Session. Discussion of problems 6 and 8 of Homework 2.

27. October 7th, 2014. Invited speaker, Professor Jorge Mario Ramirez Osorio on applications of graphs to cutting-edge problems of hydrology, social sciences en biology.

28. October 9th, 2014. (Lecture in English) Theorem 15 page 22, Modern Graph Theory. Béla Bollobás. Corollaries 9.5.2, 9.5.3, 9.5.4 and 9.5.5 pages 144-145, Graph Theory With Applications. Bondy & Murty.

29. October 13th, 2014. Debate Session. Suspended due to holiday.

30. October 14th, 2014. The Course starts being fully lectured in English. Graph Isomorphisms, Section 1.2. Graph Theory With Applications. Bondy & Murty. Find the SECOND TEST HERE

31. October 16th, 2014. Graph Automorphisms and Homomorphisms, pags 4-7, Algebraic Graph Theory. Godsil & Royle. The adjacency matrix, Section 1.3 Graph Theory With Applications. Bondy & Murty. Homework 3 available.

32. October 20th, 2014. Debate Session. Problems 33 and 39, Homework 2.

33. October 21st, 2014. The Incidence and Adjacency Matrices, Section 1.3 Graph Theory With Applications. Bondy & Murty. Quick Review of Linear Algebra: Orthonormal Basis, Orthogonal Matrices, Eigenvalues, Eigenvectors and Eigenspaces. Applied Linear Algebra and Matrix Analysis, Thomas S. Shores.

34. October 23th, 2014. Suspended due mandatory meeting for the Lecturer.

35. October 27th, 2014. Debate Session. Discussion of problems 27 and 39.

36. October 28th, 2014. Quick Review of Linear Algebra: Permutation Matrices, Characteristic Polynomials, The eigenvalues of symmetric matrices, Orthogonality properties of the Eigenspaces for symmetric matrices, The fundamental theorem of algebra, Intuitive discussion on the Spectral theorem. Applied Linear Algebra and Matrix Analysis, Thomas S. Shores.

37. October 30th, 2014. Quick Review of Linear Algebra: Self-Adjoint operators and its representation by symmetric matrices. Proof of The Spectral theorem. Assignment on computing the impact of multiplying matrices A by permutation matrices P and its transpose. Applied Linear Algebra and Matrix Analysis, Thomas S. Shores.

38. November 3th, 2014. Debate Session, suspended due to Holiday.

39. November 4th, 2014. Analyzing the actions of permutation matrices multiplication. Lemma 8.1.1 and characteristic polynomials discussion for isomorphic graphs. Cospectral graphs, page 164 Algebraic Graph Theory. Godsil & Royle. Adjacency algebra of a graph, page 442 Introduction to Graph Theory. Douglas B. West.

40. November 6th, 2014. Lemma 8.1.2, Corollary 8.1.3, page 165 Algebraic Graph Theory. Godsil & Royle. Lemma 3.2 (INCOMPLETE), page 26 Graphs and Matrices, R.B. Bapat

41. November 10th, 2014. Debate Session. The Professor was absent due to administrative duties.

42. November 11th, 2014. Lemma 3.2, page 26 Graphs and Matrices, R.B. Bapat, page 437, 8.6.12, 438 Introduction to Graph Theory. Douglas B. West.

43. November 13th, 2014. Expositions from the participants of the course

    • Ana Bernal.

    • Andres Guillermo Vargas.

    • Jeferson Zapata.

44. November 17th, 2014. Debate Session. Suspended due to holiday.

45. November 18th, 2014. Homework 4, first draft available. Expositions from the participants of the course

    • Julián Giraldo.

    • Valentina Zapata.

46. November 20th, 2014. Theorem 8.6.14, page 439 Introduction to Graph Theory. Douglas B. West. The four fundamental spaces of a matrix: column and row space, kernel and left kernel. Orthogonality relationships. Applied Linear Algebra and Matrix Analysis, Thomas S. Shores. Introductory observations towards Theorem 8.2.1, Algebraic Graph Theory. Godsil & Royle.

47. November 24th, 2014. Debate session. Cancelled, so the participants can prepare for finals.

48. November 25th, 2014. Theorem 8.2.1, Lemma 8.2.2, Algebraic Graph Theory. Godsil & Royle.

49. November 27th, 2014. Final lecture of the course. Lemma 8.2.3, Theorem 8.3.1, Lemma 8.3.2, Algebraic Graph Theory. Godsil & Royle. Lemma 2.4, Lemma 2.5, Graphs and Matrices, R.B. Bapat. Find the THIRD TEST HERE