The book, suitable as both an introductory reference and as a text
book in the rapidly growing field of topological graph theory,
models both maps (as in map-coloring problems) and groups by means
of graph imbeddings on sufaces. Automorphism groups of both graphs
and maps are studied. In addition connections are made to other
areas of mathematics, such as hypergraphs, block designs, finite
geometries, and finite fields. There are chapters on the emerging
subfields of enumerative topological graph theory and random
topological graph theory, as well as a chapter on the composition
of English church-bell music. The latter is facilitated by
imbedding the right graph of the right group on an appropriate
surface, with suitable symmetries. Throughout the emphasis is on
Cayley maps: imbeddings of Cayley graphs for finite groups as
(possibly branched) covering projections of surface imbeddings of
loop graphs with one vertex. This is not as restrictive as it might
sound; many developments in topological graph theory involve such
imbeddings.
The approach aims to make all this interconnected material readily
accessible to a beginning graduate (or an advanced undergraduate)
student, while at the same time providing the research
mathematician with a useful reference book in topological graph
theory. The focus will be on beautiful connections, both elementary
and deep, within mathematics that can best be described by the
intuitively pleasing device of imbedding graphs of groups on
surfaces.
General
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