Ten Applications of Graph Theory

ISBN-13: 9789400971561 / Angielski / Miękka / 2011 / 264 str.

Hansjoachim Walther

Opis

Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new bran- ches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-tri- vially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics. This program, Mathematics and Its Applications, is devoted to such (new) interrelations as exempla gratia: - a central concept which plays an important role in several different mathe- matical and/or scientific specialized areas; - new applications of the results and ideas from one area of scientific endeavor into another; - influences which the results, problems and concepts of one field of enquiry have and have had on the development of another.

Recenzje

`...contains an enormous amount of material for its size, and is a useful reference work for anyone interested in operations research.'
Bulletin of the London Mathematics Society, 18 (1986)

Spis treści

0. Introduction.- 1. Flows and tensions on networks.- 1.1. Basic concepts.- 1.2. Properties of flows and tensions.- 1.3. The maximum flow problem.- 1.3.1. Introduction.- 1.3.2. The theorem of Ford and Fulkerson.- 1.3.3. Generalized theorem of Ford and Fulkerson.- 1.3.4. The multi-terminal problem.- 1.4. The maximum tension problem.- 1.4.1. The existence theorem for a tension.- 1.4.2. The problems of the shortest and the longest paths as potential problems.- 1.4.3. Algorithm for determining a shortest simple path.- 1.5. The conception of network analysis.- 1.6. Bibliography.- 2. The linear transportation problem.- 2.1. Formulation of the problem.- 2.2. The solution according to Busacker and Gowen.- 2.3. The solution according to Klein.- 2.4. Proof of minimality.- 2.5. Conclusions.- 2.6. Bibliography.- 3. The cascade algorithm.- 3.1. Formulation of the problem.- 3.2. The standard method.- 3.3. The revised matrix algorithm.- 3.4. The cascade algorithm.- 3.5. Bibliography.- 4. Nonlinear transportation problems.- 4.1. Formulation of the problem.- 4.2. A convex transportation problem.- 4.3. A multi-flow problem.- 4.4. Bibliography.- 5. Communication and supply networks.- 5.1. Formulation of the problem.- 5.2. Networks without Steiner’s points.- 5.3. Networks containing Steiner’s points.- 5.4. Influence exerted by the cost function on the structure of the optimal network.- 5.5. Bibliography.- 6. The assignment and the travelling salesman problems.- 6.1. The assignment problem.- 6.1.1. Formulation of the problem.- 6.1.2. A solution algorithm for the assignment problem.- 6.2. The travelling salesman problem.- 6.2.1. Formulation of the problem.- 6.2.2. A branch-and-bound solution algorithm for the travelling salesman problem.- 6.2.3. A heuristic method for solving the travelling salesman problem.- 6.3. Final observations.- 6.4. Bibliography.- 7. Coding and decision graphs.- 7.1. Formulation of the problem.- 7.2. Algorithm for the generation of a cycle-free questionnaire.- 7.3. Optimal questionnaires.- 7.4. An example from coding.- 7.5. Bibliography.- 8. Signal flow graphs.- 8.1. Formulation of the problem.- 8.2. The algorithm of Mason for solving linear systems of equations.- 8.3. Bibliography.- 9. Minimum sets of feedback arcs.- 9.1. Formulation of the problem.- 9.2. The algorithm of Lempel and Cederbaum.- 9.3. The idea of Younger.- 9.4. Bibliography.- 10. Embedding of planar graphs in the plane.- 10.1. Formulation of the problem.- 10.2. Theorems of Kuratowski, MacLane and Whitney.- 10.3. The planarity algorithm of Dambitis.- 10.4. Planarity studies made by decomposing graphs.- 10.5. The embedding algorithm of Demoucron, Malgrange and Pertuiset.- 10.6. The planarity algorithm of Tutte.- 10.7. Bibliography.- Algorithms.- Author Index.