AUTOMORPHISMS OF NEIGHBORHOOD SEQUENCE OF A GRAPH

Abstract

Let $G$ be a graph, $u$ be a vertex of $G$, and $B(u)$ (or $B_G(u)$) be the set of $u$ with all its neighbors in $G$. A sequence $(B_1,B_2,...,B_n)$ 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 $(v_1,v_2,...,v_n)$ of $S$ such that $B(v_i)=B_i$ for $i=1,2,...,n$. Define $Aut(B_1,B_2,...,B_n)$ as theset $\{f: f$ is a permutation of $V(G)$ and $(f(B_1),f(B_2),...,f(B_n))$ is a permutation of $B_1$, $B_2$, ...,$B_n\}$. In this paper, we first prove that, for every finite group $\Gamma$, there exists a neighborhood sequence $(B_1,B_2,...,B_n)$ such that $\Gamma$ is group isomorphic to $Aut(B_1,B_2,...,B_n)$. Second, we show that, for each finite group $\Gamma$, there exists a neighborhood sequence $(B_1,B_2,...,B_n)$ such that, for eachsubgroup $H$ of $\Gamma$, $H$ is group isomorphic to $Aut(E_1,E_2,...,E_t)$ for some neighborhood sequence $(E_1,E_2,...,E_t)$ where $E_i \subseteq B_{j_i}$ and $j_1 \lt j_2 \lt \cdots \lt j_t$. Finally, we give some classes of graphs $G$ withneighborhood sequence $(B_1,B_2,...,B_n)$ satisfying that $Aut(G)$ and $Aut(B_1,B_2,...,B_n)$ are different.