Graph theory results are widely used to model and solve many engineering, social and natural science problem; it is also an excellent mean to explore proof techniques in discrete mathematics. This course aims at introducing basic graph theory concepts, and it demonstrates how they can be used in facing real-life modeling and design problem.
The main goal of this course is to equip the students with graph theory “state of mind” in facing engineering problems. The students will acquire graph theory basic knowledge and will experiencing solutions to some common problems, which will direct them towards utilizing analytical approach in their R&D challenges, in addition to simulation and experiments, which are commonly used in R&D.
Graph theory, Algorithms, Complexity, Linear algebra, Combinatorics, and Probability
This course aims at introducing basic graph theory concepts, and it demonstrates how they can be used in facing real-life modeling and design problem. Its approach it a mix of formal and intuition, where no previous knowledge in graph theory is assumed. There will be formal proofs of some important theorems (though few), while others will only be overviewed. Algorithms and complexity will only be briefly discussed as those are widely covered in other courses. Each of the topics will demonstrate related practical problems.
|第1回||Introduction||representations, isomorphism, graph structures, trees, flows, connectivity, transitivity, 3-connected graphs.|
|第2回||Matching||maximum matching, bipartite graphs, perfect matching, matching algorithms.|
|第3回||Graph coloring||vertex coloring, the chromatic number, perfect graphs, map coloring, edge coloring.|
|第4回||Connectivity||vertex connectivity, edge connectivity, 3-connected graphs.|
|第5回||The probabilistic method||random graphs, expectation, variance, evolution of random graphs.|
|第6回||Planar graphs||Jordan curve, duality, Euler formula, bridges, planarity recognition, the four-color problem.|
|第7回||Graphs and matrices||adjacency and incidence, eigenvectors, ranks, symmetric graphs.|
|第8回||Electrical networks||circulations and tensions, the matrix-tree theorem, resistive electrical networks, perfect squares, random walks on graphs.|
All lectures slides will be available on-line.
J.A. Bondy and U.S.R. Murty, Graph Theory, Springer.
D.B. West, Introduction to Graph Theory, Prectice-Hall.
Learning achievement is evaluated by the quality of the written reports, exercise problems, and etc.
Students are supposed to have some background in algorithms, basic knowledge of linear algebra, combinatorics and probability.
atsushi [at] ict.e.titech.ac.jp
Contact by e-mail in advance to schedule an appointment