Graph Symmetry: Algebraic Methods and Applications
Gena Hahn, Gert Sabidussi
Springer Science & Business Media, Jun 30, 1997 - Mathematics - 418 pages
The last decade has seen two parallel developments, one in computer science, the other in mathematics, both dealing with the same kind of combinatorial structures: networks with strong symmetry properties or, in graph-theoretical language, vertex-transitive graphs, in particular their prototypical examples, Cayley graphs. In the design of large interconnection networks it was realised that many of the most fre quently used models for such networks are Cayley graphs of various well-known groups. This has spawned a considerable amount of activity in the study of the combinatorial properties of such graphs. A number of symposia and congresses (such as the bi-annual IWIN, starting in 1991) bear witness to the interest of the computer science community in this subject. On the mathematical side, and independently of any interest in applications, progress in group theory has made it possible to make a realistic attempt at a complete description of vertex-transitive graphs. The classification of the finite simple groups has played an important role in this respect.
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Cayley graphs and interconnection networks
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2-arc transitive abelian adjacent algebra algorithms Aut(X automorphism group bipartite graph blocks cartesian product Cay(G,S Cayley graphs chromatic number circulant graphs colouring combinatorial complete graph Comput conjecture connected graph construction contains Corollary coset countable cycle defined Definition denote diameter digraph digraph F Discrete Math disjoint distance transitive graphs double ray edge edge-transitive eigenvalues element embedding equivalent example exists finite graphs function G and H graph G graph homomorphisms Graph Theory graphs of order group G Hence homogeneous homomorphism hypercube implies imprimitive induced subgraph infinite integer isomorphic Kneser graphs Lemma length Let F Let G linear locally finite matrix metacirculant orbital graphs pairs partition path Petersen graph polynomial prime primitive problem product of graphs Proof Let Proposition quasiprimitive quotient regular graphs relation result retracts Section self-paired semidefinite programming simple group structure subset symmetric Theorem transposition valency vertex vertex set vertex-transitive graphs VT-graphs wreath product