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Zero-symmetric graphs. Trivalent graphical regular representations of groups. (English) Zbl 0548.05031

New York etc.: Academic Press. IX, 170 p. $ 19.50 (1981).
The book is devoted to the study of trivalent graphs whose automorphism group acts regularly on the vertices. Such graphs are called 0-symmetric graphs. The authors describe and use a variety of methods for obtaining 0-symmetric graphs in an attempt to compile a census of those having not more than 120 vertices. Every 0-symmetric graph is the Cayley graph of its automorphism group with respect to either 3 involutory generators (class \({}^ 3Z)\) or to one involutory and one non-involutory generator (class \({}^ 1Z)\). The main method is to start with a family of groups together with generators and to check which of these groups have 0- symmetric Cayley graphs. The authors give necessary conditions for a group to have a 0-symmetric Cayley graph, but they have found sufficient conditions only for special families of groups. In other cases they had to check individually each graph in order to be sure that the graph has no ”hidden symmetry”. Most of these checks are not carried out in the book. All known 0-symmetric graphs with at most 120 vertices are listed in tables. In table 22.1 are listed (from an unpublished work of R. M. Foster) the 350 of these graphs which are Cayley graphs of dihedral groups with respect to three involutory generators. In tables at the end of the book are listed 90 further graphs of type \({}^ 3Z\) and 28 graphs of type \({}^ 1Z\). For each of these graphs is given its girth, a name for its automorphism group, the reference where in the book the graph can be found and the LCF code for those graphs for which a Hamiltonian circuit has been found, which are all except 8 of the graphs.
Reviewer: U.Brehm

MSC:

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05-02 Research exposition (monographs, survey articles) pertaining to combinatorics
20F05 Generators, relations, and presentations of groups
20F65 Geometric group theory