## Taiwanese Journal of Mathematics

### AUTOMORPHISMS OF NEIGHBORHOOD SEQUENCE OF A GRAPH

Li-Da Tong

#### 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.

#### Article information

Source
Taiwanese J. Math., Volume 19, Number 4 (2015), 1085-1096.

Dates
First available in Project Euclid: 4 July 2017

https://projecteuclid.org/euclid.twjm/1499133690

Digital Object Identifier
doi:10.11650/tjm.19.2015.4816

Mathematical Reviews number (MathSciNet)
MR3384680

Zentralblatt MATH identifier
1357.05060

#### Citation

Tong, Li-Da. AUTOMORPHISMS OF NEIGHBORHOOD SEQUENCE OF A GRAPH. Taiwanese J. Math. 19 (2015), no. 4, 1085--1096. doi:10.11650/tjm.19.2015.4816. https://projecteuclid.org/euclid.twjm/1499133690

#### References

• D.-B. Chang and L.-D. Tong, Choice identification of a graph, Discrete Applied Mathematics, accepted 2013.
• R. Frucht, Herstellung von Graphen mit vorgegebener abstrakter Gruppe, Compositio Mathematica (in German), 6 (1939), 239-250.
• S. Gravier, J. Moncel and A. Semri, Identifying codes of cycles, European J. of Combin., 27 (2006), 767-776.
• M. G. Karpovsky, K. Chakrabarty and L. B. Levitin, On a new class of codes for identifying vertices in graphs, IEEE Transactions on Information Theory, 44(2) (1998), 599-611.
• M. G. Karpovsky, K. Chakrabarty, L. B. Levitin and D. R. Avreky, On the covering of vertices for fault diagnosis in hypercubes, Inform. Process. Lett., 69 (1999), 99-103.
• A. Raspaud and L.-D. Tong, Minimum identifying code graphs, Discrete Applied Mathematics, 160(9) (2012), 1385-1389.
• M. Xu, K. Thulasiramanb and X. Hu, Identifying codes of cycles with odd orders, European J. of Combin., 29 (2008), 1717-1720.