Graphs and Geometry

(a.k.a. Signed Graphs and their Friends)

Fall 2008

Spring-Summer 2009

Spring 2010

- Here are the current
**Course Notes PDF | Course Notes PS**. - Topically arranged course notes PDF | Topically arranged course notes PS, as of Oct. 10; good for Chapters O, I.
- Here is the current course outline.
- Here are the recommended readings.
- Here is the
**note-taking schedule**. - Here is the Tex-files access page.
- Here are guidelines for LaTeXing the course notes.
- Fraktur (German) letters, printed and handwritten forms, from Sheila MacIntyre and Edith Witte, German-English Mathematical Vocabulary, Oliver and Boyd, Edinburgh and London, 1956; second ed. 1966. Higher resolution (2 MB).

This course is an introduction to fundamental concepts of signed graph theory, generalizations, and related geometry, including a dash of matroids. There is no official book but there are lots of papers available (electronically, and I can give real reprints of several of them to those who like that sort of thing). A linked list will be posted later. I'll recommend a basic graph theory book when I can think of one that's most appropriate. (Diestel or Tutte will be satisfactory though not perfect.)

The course is not normally suitable for first-year graduate students. However, there are **no specific prerequisites** except the traditional ``mathematical maturity''—not even a course in graph or matroid theory, though it will be helpful. Basic graduate algebra is *very* helpful.

**SPRING 2010**:

**Tu 12:00 - 1:05**in LN-2205,**Th 1:15 - 2:40 or earlier**in LN-2206.

I will be happy to see graduate students at any time, as far as possible (that means, not early in the morning and not when I'm rushing to prepare a class). We can make appointments, but they're not required.

All students are *strongly encouraged*, if not required, to attend the Combinatorics Seminar, usually on Tuesdays, 1:15 - 2:15. There will be talks you can't understand, as well as some you can't help understanding, on all kinds of topics in graph theory and other combinatorics as well as in number theory (and sometimes both at once). (You'll be surprised how much you learn by not understanding a great many talks.)

I will lecture, for the most part.

Student work: You will have to work hard sometimes to understand the lectures and readings. Keep your colored pencils handy. Ask questions!

I will have you write up the notes of the lectures (in LaTeX). I'll use them to prepare lecture notes.

- Here are the course notes at this moment (not necessarily final):
**Course Notes PDF | Course Notes PS**. - Here are the topically arranged course notes in PDF, or in PS, as of Oct. 10. They will not be kept up to date (until all the daily reports are in, after the semester), but they're nearly complete for Chapters O, I.

Link to the

Your grade will be based partly on your note-taking and write-ups. It may be based partly on student presentations. I will also meet with each of you individually on a regular basis.

Here is the **Tex-files page** from which you can download the Latex files of all the lectures.

For the complete list see the Readings page. Here is the most basic list.

- [Cox] H.S.M. Coxeter, Introduction to Geometry, second edition. Chapters 12–14, §§ 15.1-3. Basic geometrical background for use in the course.
- [Pearls] Nora Hartsfield and Gerhart Ringel, Pearls in Graph Theory. Ch. 1, §§ 2.1-2, § 3.1, Ch. 8 for elementary background in graph theory. Very readable. Not always precisely correct, so make sure you understand the proofs.

- [SG] T.Z., "Signed graphs". Discrete Appl. Math., 4 (1982), 47-74 (and correction). Basic (but not necessarily elementary) concepts and properties.
- [SGC] T.Z., "Signed graph coloring". Discrete Math., 39 (1982), 215-228. Basic (but not necessarily elementary) concepts and properties of coloring and the chromatic polynomial.
- [MTS] T.Z., "Matrices in the theory of signed simple graphs". Introductory survey.

- [GRS] T.Z., "The geometry of root systems and signed graphs". Amer. Math. Monthly, 88 (Feb., 1981), no. 2, 88-105. A readable introduction to some of the connections between graph theory and geometry.

- [BG1] T.Z., "Biased graphs. I. Bias, balance and gains". Fundamentals of gain graphs and biased graphs.
- [BG2] T.Z., "Biased graphs. II. The three matroids". The closure operations that are basic to the theory, in §§ 2, 3.

- Here are the current
**Course Notes PDF | Course Notes PS**. - Topically arranged course notes PDF | Topically arranged course notes PS, as of Oct. 10; good for Chapters O, I.
- Here is the current course outline.
- Here are the recommended readings.
- Here is the note-taking schedule.
- Here are guidelines for LaTeXing the course notes.

To my home page.