GRAPH THEORY

The couse has the following objectives:

Knowledge and understanding:

- Mathematic Instruments in graph theory such as theorems and algoritms will be provide in the course. They permit to develop mathematical abilities in reasoning and calculation. These abilities could permit to resolve known problems by mathematical model.

Applying and knowledge and understanding:

- At the end of course student will be able to get knowledge for a tightened use of new mathematical techniques and an understanding of treated arguments in such way to link them each other.

Making judgements:

- Course is based on logical-deductive method which wants to give to students authonomus judgement useful to understanding incorrect method of demonstration also, by logical reasoning, student will be able to face not difficult problems in graph theory with teacher's help.

Communication skills:

- In the final exam, student must show for learned different mathematical techniques an adapt maturity on oral communication using also multimedia tools.

Learning skills

- Autonomously student will be able to face application and theoretical arguments which could be studied in new classes or in different working fields; for example flow theory and connectivity have huge application on telecommunications field (Local Area Network and Metropolitan Area Network: LAN e MAN), on electrical and communication fields (Industrial design).

The lessons will be held through classroom. In these lessons the program will be divided into the following sections: basic notions, planar graphs, cycles and cocycles, different graph connections, graph fluxes, matchings and coverings, colorability.

In each of these sections first it will be discussed the main theoretical topics and then showed how these topics can be linked to possible applications. Then algorithms can be presented, and they allow in many cases to identify particular graphs or solutions proposed by the theoretical results

Basic definition on Graph Theory. Cycles and cocycles. Cyclomatic and cocyclomatic numbers. Planar graph and their property. Euler's formula. Tree and cotree. Spanning tree. Strongly and minimally connected graphs and their property. Maximum and perfect matchings. Covering and minimum covering of a graph. Matching in bipartite graph, Köenig's theorem. Hamiltonian and Eulerian graphs. Edge and vertex colourings, chromatic number and index number. Vizing's theorem. Basic notion on Hypergrapghs and G-designs. Properties of digraphs.

1. C. Berge, "Graph and Hypergraph", Elsevier.
2. M. Gionfriddo, Notes on Graph Theory 2017
3. D. West, Introduction to Graph Theory
4. V. Voloshin, Introduction to Graph Theory