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This book introduces oriented version of metrics and cuts and their
multidimensional analogues, as well as partial metrics and weighted
metrics. It is a follow-up of Geometry of Cuts and Metrics by Deza
and Laurent which presents rich theory of classical binary and
symmetric objects - metrics and cuts.Many research publications on
this subject are devoted to different special aspects of the theory
of generalized metrics. However, they are disconnected one from
other, often written in different mathematical language, consider
the same objects from different points of view without analysis of
possible connections, etc. In this book we will construct full
theory of main classes of finite generalized metrics and their
polyhedral aspects.
In this volume very simplified models are introduced to understand
the random sequential packing models mathematically. The
1-dimensional model is sometimes called the Parking Problem, which
is known by the pioneering works by Flory (1939), Renyi (1958),
Dvoretzky and Robbins (1962). To obtain a 1-dimensional packing
density, distribution of the minimum of gaps, etc., the classical
analysis has to be studied. The packing density of the general
multi-dimensional random sequential packing of cubes (hypercubes)
makes a well-known unsolved problem. The experimental analysis is
usually applied to the problem. This book introduces simplified
multi-dimensional models of cubes and torus, which keep the
character of the original general model, and introduces a
combinatorial analysis for combinatorial modelings.
The central theme of the present book is zigzags and
central-circuits of three- or four-regular plane graphs, which
allow a double covering or covering of the edgeset to be obtained.
The book presents zigzag and central circuit structures of
geometric fullerenes and several other classes of graph of interest
in the fields of chemistry and mathematics. It also discusses the
symmetries, parameterization and the Goldberg-Coxeter construction
for those graphs. It is the first book on this subject, presenting
full structure theory of such graphs. While many previous
publications only addressed particular questions about selected
graphs, this book is based on numerous computations and presents
extensive data (tables and figures), as well as algorithmic and
computational information. It will be of interest to researchers
and students of discrete geometry, mathematical chemistry and
combinatorics, as well as to lay mathematicians.
Polycycles and symmetric polyhedra appear as generalisations of
graphs in the modelling of molecular structures, such as the Nobel
prize winning fullerenes, occurring in chemistry and
crystallography. The chemistry has inspired and informed many
interesting questions in mathematics and computer science, which in
turn have suggested directions for synthesis of molecules. Here the
authors give access to new results in the theory of polycycles and
two-faced maps together with the relevant background material and
mathematical tools for their study. Organised so that, after
reading the introductory chapter, each chapter can be read
independently from the others, the book should be accessible to
researchers and students in graph theory, discrete geometry, and
combinatorics, as well as to those in more applied areas such as
mathematical chemistry and crystallography. Many of the results in
the subject require the use of computer enumeration; the
corresponding programs are available from the author's website.
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