Elements of graphing data by William S. Cleveland

By William S. Cleveland

Visualization equipment in technological know-how and
technology. Many new principles and methods;
many no longer well known sooner than. Excellent
methodological source for research

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If a random variable x has any of n possible values that are equally probable, then one says that x is discretely uniformly distributed. 14 Let v be a vertex of a vertex-transitive graph G, and ϕ1 , ϕ2 , . . , ϕk be uniformly distributed, randomly and independently chosen automorphisms. Then the vertices ϕ1 (v), ϕ2 (v), . . , ϕk (v) are also uniformly distributed. Proof We first consider two automorphisms ϕ, ψ that map v into u. Then ψ −1 ϕ(v) = v, hence ψ −1 ϕ is in the stabilizer Aut(G)v of v.

Vj , . . , ur ) → (v1 , v2 , . . , vj , . . , vi , . . , vr ) is an automorphism of Qr . Furthermore, for any i, 1 ≤ i ≤ r, ψi : (v1 , v2 , . . , vi , . . , vr ) → (v1 , v2 , . . , vi + 1, . . , vr ) , where addition is modulo 2, is also an automorphism. Clearly, id = ψi2 for all i, and ψi ψj = ψj ψi for all 1 ≤ i, j ≤ r. Hence, the subgroup of Aut(Qr ) generated by the ψi is Abelian, and each nontrivial element has order 2. Such groups are known as elementary Abelian 2-groups and also called Boolean.

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 37 40 41 41 43 44 46 We now introduce the primary object of interest in this book: the idea of a graph product. Broadly speaking, a graph product is a binary operation on Γ or Γ0 . However, under reasonable and natural restrictions (such as associativity), the number of different products is actually quite limited.

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