MATH32091 – 2019/20

Specification

Aims

To introduce the students to graphs, their properties and their applications as models of networks.
To introduce the students to generating functions and their applications.

Brief Description of the unit

A graph consists of a set of vertices with a set of edges connecting some pairs of vertices. Depending on the context, the edges may represent a mathematical relation, two people knowing each other or roads connecting towns, etc. The graph theory part of the course deals with networks, structure of graphs, and extremal problems involving graphs.

The combinatorial half of this course is concerned with enumeration, that is, given a family of problems P(n), n a natural number, find a(n), the number of solutions of P(n) for each such n. The basic device is the generating function, a function F(t) that can be found directly from a description of the problem and for which there exists an expansion in the form F(t) = sum {a(n)g_n(t); n a natural number}. Generating functions are also used to prove a family of counting formulae, to prove combinatorial identities and to obtain asymptotic formulae for a(n).

Learning Outcomes

• define graph theoretic concepts, state and prove their properties,
• describe graph theoretic algorithms and prove their correctness,
• formulate problems in terms of graphs and apply the theorems and algorithms taught in the course to solve them,
• define the various types of generating functions,
• state and prove the basic properties of generating functions,
• use generating functions to solve a variety of combinatorial problems.

Syllabus

Graph Theory
1. Introduction [1 lecture]
2. Electrical networks [2]
3. Flows in graphs, Max-flow min-cut theorem [3]
4. Matching problems [3]
5. Extremal problems [3]

Combinatorics
1. Examples using ordinary power series and exponential generating functions, general properties of such functions. [3]
2. Dirichlet Series as generating functions. [1]
3. A general family of problems described in terms of "cards, decks and hands" with solution methods using generating functions. [3]
4. Generating function proofs of the sieve formula and of various combinatorial identities. Certifying combinatorial identities. [2]
5. Some analytical methods and asymptotic results. [2]

Textbooks

• B Bollobás, Graph theory, An introductory course (Graduate Texts in Mathematics 63), Springer, 1979.
• B Bollobás, Modern graph theory, (Graduate Texts in Mathematics 184), Springer, 1998.
• D Jungnickel, Graphs, Networks and Algorithms (Algorithms & Computation in Mathematics 5), Springer, 1998.
• H S Wilf, generatingfunctionology, A K Peters, 3rd ed., 2006.

Teaching and learning methods

Two lectures and one examples class each week. Students are recommended to do at least four hours private study each week on the course unit.

Assessment