组合数学和优化数学硕士[合作社]-图论

图由一组元素以及在该集合上定义的二进制关系组成。图可以用图表表示，其中元素显示为点，二进制关系显示为连接点对的线。正是这种表示法使图论得到了它的名称和很大的吸引力。然而
Combinatorics is the study of discrete structures and their properties. Many modern scientific advances have employed combinatorial structures to model the physical world, and recent advances in computational technology have made such investigations feasible. In particular, since computers process discrete data, combinatorics has become indispensable to computer science. Optimization, or mathematical programming, is the study of maximizing and minimizing functions subject to specified boundary conditions or constraints. With the emergence of computers, optimization experienced a dramatic growth as a mathematical theory, enhancing both combinatorics and classical analysis. The functions to be optimized arise in engineering, the physical and management sciences, and in various branches of mathematics. In the co-op option, students spend one or more terms working in industry, thus gaining valuable experience while earning their degree. Recently, the co-op option has been undertaken by students whose research focuses on financial applications of optimization. It is recommended that students interested in this option apply for the ordinary (non-co-op) MMath program and then contact the graduate officer about transferring to the co-op program subsequent to their admission.<br><br>A graph consists of a set of elements together with a binary relation defined on the set. Graphs can be represented by diagrams in which the elements are shown as points and the binary relation as lines joining pairs of points. It is this representation which gives graph theory its name and much of its appeal. However, the true importance of graphs is that, as basic mathematical structures, they arise in diverse contexts, both theoretical and applied. The concept of a graph was known already to Euler in the early eighteenth century, but it was the notorious Four-Colour Problem, posed by F. Guthrie in the mid-nineteenth century, that spurred the development of this simple concept into a flourishing theory. In this century, interactions between graph theory and linear algebra, probability theory, number theory, group theory, geometry, topology, and other branches of mathematics have led to further developments in the subject. In recent years, its fundamental links with operations research and computer science have resulted in the fast growth and greatly increased prominence of graph theory.