This book gives an elementary treatment of the basic material about graph spectra, both for ordinary, and Laplace and Seidel spectra. The text progresses systematically, by covering standard topics before presenting some new material on trees, strongly regular graphs, two-graphs, association schemes, p-ranks of configurations and similar topics. Exercises at the end of each chapter provide practice and vary from easy yet interesting applications of the treated theory, to little excursions into related topics. Tables, references at the end of the book, an author and subject index enrich the text. Spectra of Graphs is written for researchers, teachers and graduate students interested in graph spectra. The reader is assumed to be familiar with basic linear algebra and eigenvalues, although some more advanced topics in linear algebra, like the Perron-Frobenius theorem and eigenvalue interlacing are included.

Spectra of graphs

Introduction to linear and nonlinear programming

The theory of directed graphs has developed enormously over recent decades, yet this book (first published in 2000) remains the only book to cover more than a small fraction of the results. New research in the field has made a second edition a necessity. Substantially revised, reorganised and updated, the book now comprises eighteen chapters, carefully arranged in a straightforward and logical manner, with many new results and open problems. As well as covering the theoretical aspects of the subject, with detailed proofs of many important results, the authors present a number of algorithms, and whole chapters are devoted to topics such as branchings, feedback arc and vertex sets, connectivity augmentations, sparse subdigraphs with prescribed connectivity, and also packing, covering and decompositions of digraphs. Throughout the book, there is a strong focus on applications which include quantum mechanics, bioinformatics, embedded computing, and the travelling salesman problem. Detailed indices and topic-oriented chapters ease navigation, and more than 650 exercises, 170 figures and 150 open problems are included to help immerse the reader in all aspects of the subject. Digraphs is an essential, comprehensive reference for undergraduate and graduate students, and researchers in mathematics, operations research and computer science. It will also prove invaluable to specialists in related areas, such as meteorology, physics and computational biology.

Digraphs Theory Algorithms And Applications

Mathematical programming models with noisy, erroneous, or incomplete data are common in operations research applications. Difficulties with such data are typically dealt with reactively-through sensitivity analysis-or proactively-through stochastic programming formulations. In this paper, we characterize the desirable properties of a solution to models, when the problem data are described by a set of scenarios for their value, instead of using point estimates. A solution to an optimization model is defined as: solution robust if it remains "close" to optimal for all scenarios of the input data, and model robust if it remains "almost" feasible for all data scenarios. We then develop a general model formulation, called robust optimization RO, that explicitly incorporates the conflicting objectives of solution and model robustness. Robust optimization is compared with the traditional approaches of sensitivity analysis and stochastic linear programming. The classical diet problem illustrates the issues. Robust optimization models are then developed for several real-world applications: power capacity expansion; matrix balancing and image reconstruction; air-force airline scheduling; scenario immunization for financial planning; and minimum weight structural design. We also comment on the suitability of parallel and distributed computer architectures for the solution of robust optimization models.

Robust Optimization of Large-Scale Systems

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Automatic discovery of linear restraints among variables of a program

https://www.tayfunsonmez.net/wp-content/uploads/2013/10/Balinski-Sonmez-JET991.pdf

A Tale of Two Mechanisms: Student Placement

The issue of fair representation will take center stage as U.S. congressional districts are reapportioned based on the 2000 Census. Using U.S. history as a guide, the authors develop a theory of fair representation that establishes various principles for translating state populations --or vote totals of parties --into a fair allocation of congressional seats. They conclude that the current apportionment formula cheats the larger states in favor of the smaller, contrary to the intentions of the founding fathers and compromising the Supreme Court's ""one man, one vote"" rulings. Balinski and Young interweave the theoretical development with a rich historical account of controversies over representation, and show how many of these principles grew out of political contests in the course of United States history. The result is a work that is at once history, politics, and popular science. The book --updated with data from the 1980 and 1990 Census counts --vividly demonstrates that apportionment deals with the very substance of political power.

Fair Representation: Meeting the Ideal of One Man, One Vote

An actual truck delivery problem of general applicability is stated as an integer program. The successful computational performance of Gomory's "cutting plane" algorithm for a set of nine particular problems is discussed. Finally, the structure of the general problem is shown to be a natural generalization of covering problems of graph theory.

On an Integer Program for a Delivery Problem

The issue of fair representation will take center stage as U.S. congressional districts are reapportioned based on the 2000 Census. Using U.S. history as a guide, the authors develop a theory of fair representation that establishes various principles for translating state populations --or vote totals of parties --into a fair allocation of congressional seats. They conclude that the current apportionment formula cheats the larger states in favor of the smaller, contrary to the intentions of the founding fathers and compromising the Supreme Court's \"\"one man, one vote\"\" rulings. Balinski and Young interweave the theoretical development with a rich historical account of controversies over representation, and show how many of these principles grew out of political contests in the course of United States history. The result is a work that is at once history, politics, and popular science. The book --updated with data from the 1980 and 1990 Census counts --vividly demonstrates that apportionment deals with the very substance of political power.

Fair Representation: Meeting the Ideal of One Man, One Vote.

This paper formulates a fixed-cost transportation problem as an integer program, describes some of its special properties, and suggests an approximate method of solution. Examples are given to demonstrate the approximation technique.

Michel Balinski

Papers

A Tale of Two Mechanisms: Student Placement

Fair Representation: Meeting the Ideal of One Man, One Vote

On an Integer Program for a Delivery Problem

Fair Representation: Meeting the Ideal of One Man, One Vote.

Fixed‐cost transportation problems