On the Automorphism Group of Token Graphs of Complete Bipartite Graphs

TL;DR

This paper determines the automorphism group of the $k$-token graph derived from complete bipartite graphs, providing insights into their symmetry properties and structural automorphisms.

Contribution

It explicitly characterizes the automorphism group of the $k$-token graph of $K_{m,n}$, a problem previously unresolved.

Findings

Automorphism group of $k$-token graphs of $K_{m,n}$ is explicitly determined.

Results reveal symmetry structures depend on the bipartite partition sizes.

Provides foundational understanding for symmetry analysis in token graphs.

Abstract

Let $G$ be a graph of order $n$ and let $k\in \{1,2,\ldots,n-1\}$. The $k$-token graph of $G$ is the graph, whose vertices are all the $k$-subsets of vertices of $G$, where two such $k$-sets are adjacent whenever their symmetric difference is an edge of $G$. In this paper, we determine the automorphism group of the $k$-token graph of the complete bipartite graph $K_{m,n}$.