Graphs, Dioids and Semirings New Models and Algorithms [Paperback]

The primary objective of this essential text is to emphasize the deep relations existing between the semiring and dio?d structures with graphs and their combinatorial properties. It does so at the same time as demonstrating the modeling and problem-solving flexibility of these structures. In addition the book provides an extensive overview of the mathematical properties employed by nonclassical algebraic structures which either extend usual algebra or form a new branch of it.

Pre-Semirings, Semirings and Dioids.- Combinatorial Properties of (Pre)-Semirings.- Topology on Ordered Sets: Topological Dioids.- Solving Linear Systems in Dioids.- Linear Dependence and Independence in Semi-Modules and Moduloids.- Eigenvalues and Eigenvectors of Endomorphisms.- Dioids and Nonlinear Analysis.- Collected Examples of Monoids, (Pre)-Semirings and Dioids.

From the reviews:

The authors carefully explain with examples what is meant by a canonically ordered monoid, semiring and dioid, and go on to show in the book the relevance of these algebraic structures to such classic operations research problems related to graphs and networks & . recommend this as a book to be added in the libraries of institutions where graduate courses in OR are taught, because it may provide ideas for PhD students and others to explore further the concepts developed in this book & . (R Bharath, Journal of the Operational Research Society, Vol. 60, 2009)

The origins of Graph Theory date back to Euler (1736) with the solution of the celebrated 'Koenigsberg Bridges Problem'; and to Hamilton with the famous 'Trip around the World' game (1859), stating for the first time a problem which, in its most recent version  the 'Traveling Salesman Problem' -, is still the subject of active research. Yet, it has been during the last fifty years or sowith the rise of the electronic computersthat Graph theory has become an indispensable dlCa