Automorphisms of Neighborhood Sequence of a Graph
نویسنده
چکیده
Let G be a graph, u be a vertex of G, and B(u)(or BG(u)) be the set of u with all its neighbors in G. A sequence (B1, B2, ..., Bn) of subsets of an n-set S is a neighborhood sequence if there exists a graph G with a vertex set S and a permutation (v1, v2, ..., vn) of S such that B(vi) = Bi for i = 1, 2, ..., n. Define Aut(B1, B2, ..., Bn) as the set {f : f is a permutation of V (G) and (f(B1), f(B2), ..., f(Bn)) is a permutation of B1, B2, ..., Bn}. In this paper, we first prove that, for every finite group Γ, there exists a neighborhood sequence (B1, B2, ..., Bn) such that Γ is group isomorphic to Aut(B1, B2, ..., Bn). Second, we show that, for each finite group Γ, there exists a neighborhood sequence (B1, B2, ..., Bn) such that, for each subgroup H of Γ, H is group isomorphic to Aut(E1, E2, .., Et) for some neighborhood sequence (E1, E2, .., Et) where Ei ⊆ Bji and j1 < j2 < ... < jt. Finally, we give some classes of graphs G with neighborhood sequence (B1, B2, ..., Bn) satisfying that Aut(G) and Aut(B1, B2, ..., Bn) are different.
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