This book contains the following chapters in company with related subjects. In Chapter I, the embedding of a graph on surfaces are much
concerned because they are motivated to building up the theo
ry of abstract maps related with Smarandache geometry. The second chapter is for the formal definition of abstract maps. One can see that this matter is a natural generalization of gr
aph embedding on surfaces.
The third chapter is on the duality not only for maps themselves
but also for operations on maps from one surface to another. One
can see how the duality is naturally deduced from the abstract maps
described in the second chapter.
The fourth chapter is on the orientability. One can see how the
orientability is formally designed as a combinatorial invariant. The
fifth chapter concentrates on the classification of orientable maps. The
sixth chapter is for the classification of nonorientable maps.
From the two chapters: Chapter V and Chapter VI, one can see how the procedure is simplified for these classifications.
The seventh chapter is on the isomorphisms of maps and pro-
vides an efficient algorithm for the justification and recogni
tion of the isomorphism of two maps, which has been shown to be useful for
determining the automorphism group of a map in the eighth chapter.
Moreover, it enables us to access an automorphism of a graph.
The ninth and the tenth chapters observe the number of distinct
asymmetric maps with the size as a parameter. In the former, only
one vertex maps are counted by favorite formulas and in the latter,
general maps are counted from differential equations. .
The next chapter, Chapter XI, only presents some ideas for accessing the symmetric census of maps and further, of graphs. This
topic is being developed in some other directions and left as
a subject written in the near future.
From Chapter XII through Chapter XV, extensions from basic theory are much concerned with further applications.
Chapter XII discusses in brief on genus polynomial of a graph and all its super maps rooted and unrooted on the basis of the joint tree model. Recent progresses on this aspect are referred to read the articles. |