Automorphism group of the graph $A(n,k,r)$

TL;DR

This paper determines the automorphism groups of the $(n,k,r)$-arrangement graphs in specific cases, resolving parts of an open problem and proposing a new conjecture in graph symmetry characterization.

Contribution

It characterizes the automorphism groups of $A(n,k,r)$ for particular parameter cases, advancing understanding of graph symmetries and addressing an open problem.

Findings

Automorphism groups characterized for $r=k$ and $r=2<k=n$ cases.

Resolved two special cases of an open problem.

Proposed a bold new conjecture on automorphism groups.

Abstract

Let $[n]^{(k)}$ be the set of all ordered $k$-tuples of distinct elements in $[n]=\{1,2,...,n\}$. The $(n,k,r)$-arrangement graph $A(n,k,r)$ with $1\leq r\leq k\leq n$, is the graph with vertex set $[n]^{(k)}$ and with two $k$-tuples are adjacent if they differ in exactly $r$ coordinates. In this manuscript, we characterize the full automorphism groups of $A(n,k,r)$ in the cases that $1\leq r=k\leq n$ and $r=2<k=n$. Thus, we resolve two special cases of an open problem proposed by Fu-Gang Yin, Yan-Quan Feng, Jin-Xin Zhou and Yu-Hong Guo. In addition, we conclude with a bold conjecture.