By H. S. M. Coxeter

Zero-Symmetric Graphs: Trivalent Graphical typical Representations of teams describes the zero-symmetric graphs with no more than one hundred twenty vertices.The graphs thought of during this textual content are finite, hooked up, vertex-transitive and trivalent.

This e-book is prepared into 3 components encompassing 25 chapters. the 1st half experiences the various periods of zero-symmetric graphs, in keeping with the variety of primarily various edges incident at each one vertex, particularly, the S, T, and Z periods. the rest elements speak about the theory and features of style 1Z and 3Z graphs. those components discover Cayley graphs of particular teams, together with the parameters of Cayley graphs of groups.

This booklet will end up invaluable to mathematicians, machine scientists, and researchers.

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**Extra resources for Zero-symmetric Graphs: Trivalent Graphical Regular Representations of Groups**

**Example text**

6) with parameters 4,3,2,3. with parameters 4,3,2,3. It be pointed pointed out It should should however however be out that that aa Cayley Cayley graph graph of of type 33s , 33T or 33 z is not always completely characterized by type S, T or Z is not always completely characterized by its four parameters. For instance, the same parameters 3,3,3,4 its four parameters. 7) and permutations in and this this latter latter group, group, consisting consisting of of the the even even permutations in the direct direct product S 4 x* S3' S 3, is is aa subgroup subgroup of of index index 2 2 of of that that the product S4 13 13 65 °5 Parameters Parameters of of graphs graphs of of type type 3 3 direct product.

1, it it is, is, however, however, easy easy to to find find aa hamiltonian hamiltonian circuit. circuit. In the the case case of of the the graph graph No. with 40 40 vertices vertices the the In No. 16) gives gives us us aa hamiltonian hamiltonian circuit circuit with with "chord "chord length length number" number" N = 33 and LCF code N and LCF code c c tion). tion) . 66 In an an analogous analogous fashion fashion we we can can use use for for the the graph graph No. 17) = E , thus thus obtaining obtaining the the LCF LCF code code [3,37,43,-3:-] [3, 37,43,-3;-] 1 3 13 ..

7) we we see see that that 3 1 2 3 ,1 2 Z(5,6,2) _ F p 3',1,2 _ F p 3 ,2,1 .. 8) 4 3 generate F 2' 1' "-1, and S 4 = (6 (6 7 7 8), 8 ) , we we have have generate P " , and S 21 1 p3,2,1 C x F 2 ,1,-1 ;; F ' ' = -= c 33 x F ' ' " that is, Z(5,6,2) Z(5,6,2) ~= C C 3 xx Z(5,2,2). z(5,2,2). 10) gives N = 44 and gives aa hamiltonian hamiltonian circuit circuit with with N and the the LCF LCF code code c c [18,9,-17,29,-9,18;-]5 [18,9,-17,29,-9,18;-]- . 12) = E RS RSRS which produces 9-gons which produces 9-gons in in the the graph, graph, and and since since no no 'shorter' 'shorter' 2 E) occurs in the group, the graph relation (apart from R = relation = E) occurs in the group, the graph (apart from R has girth 9.