By Guo L.-T.

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Vn such that first come the elements of O1 , followed by the elements of O2 and so on. Let A be the corresponding adjacency matrix of G. Let |Oi | = ni . Suppose that λ is an eigenvalue of R and let x = (x1 , . . , xr )t be a corresponding eigenvector of λ. Let x∗ be the n-vector obtained from x by consecutively repeating ni times each xi . Prove that x∗ is an eigenvector of A with eigenvalue λ. (c) Deduce that the characteristic polynomial of R divides that of A. (d) Deduce that if Aut(G) is nontrivial, then the characteristic polynomial of G is reducible over Z.

For example, α 4 β maps vertex 53 to vertex 1. However, the graph is not vertex-transitive because from an odd-numbered vertex it is possible to have three different paths of length 4 joining the vertex to some other common vertex (for example, vertex 1 to vertex 5), but this is not possible from an even-numbered vertex. Another way to show that the Gray Graph is not vertex-transitive is to consider the distance sequences of its vertices [166]. The distance sequence of a vertex v is the vector (a0 , a1 , .

1. 2. A Cayley colour graph of D4 A colour preserving automorphism of Col( , X) is a permutation π of V(Col( , X)) = such that, for all α, β ∈ , (α, β) is an arc coloured σ if and only if (π(α), π(β)) is also an arc coloured σ . The set of all colour preserving automorphisms of Col( , X) forms a group under composition of functions, and it is denoted by Aut(Col( , X)). 1 Let be a nontrivial finite group with a generating set X. Let π be a permutation of V(Col( , X)). Then π is a colour preserving automorphism of Col( , X) if and only if π(ασ ) = π(α)σ for all α ∈ and σ ∈ X.

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