Automorphism Groups of Maps, Surfaces and Smarandache Geometries (second edition), graduate textbook in mathematics

Automorphisms of a system survey its symmetry and appear nearly in all mathematical branches, such as those of algebra, combinatorics, geometry, … and theoretical physics or chemistry. The main motivation of this book is to present a systemically introduction to automorphism groups on algebra, graphs, maps, i.e., graphs on surfaces and geometrical structures with applications. Topics covered in this book include elementary groups, symmetric graphs, graphs on surfaces, regular maps, lifted automorphisms of graphs or maps, automorphisms of maps underlying a graph with applications to map enumeration, isometries on Smarandache geometry and CC conjecture, etc., which is suitable as a textbook for graduate students, and also a valuable reference for researchers in group action, graphs with groups, combinatorics with enumeration, Smarandache multispaces, particularly, Smarandache geometry with applications. The most importance of Smarandache geometries was the introduction of the degree of negation of an axiom (and more general the degree of negation of a theorem, lemma, scientific or humanistic proposition) which works somehow like the negation in fuzzy logic (with a degree of truth, and a degree of falsehood) or more general like the negation in neutrosophic logic (with a degree of truth, a degree of falsehood, and a degree of neutrality (neither true nor false, but unknown, ambiguous, indeterminate) [not only Enclid s geometrical axioms, but any scientific or humanistic proposition in any field] or partial negation of an axiom (and, in general, partial negation of a scientific or humanistic proposition in any field). These geometries connect many geometrical spaces with different structures into a heterogeneous multispace with multistructure. In general, a rule in a system is said to be Smarandachely denied if it behaves in at least two different ways within the same set, i.e. validated and invalided, or only invalided but in multiple distinct ways. A Smarandache syste