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On the Automorphism Group of Token Graphs of Complete Bipartite Graphs

Ruy Fabila-Monroy, Ana Laura Trujillo-Negrete

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 $K_{m,n}$.

On the Automorphism Group of Token Graphs of Complete Bipartite Graphs

Abstract

Let be a graph of order and let . The -token graph of is the graph, whose vertices are all the -subsets of vertices of , where two such -sets are adjacent whenever their symmetric difference is an edge of . In this paper, we determine the automorphism group of the -token graph of the complete bipartite graph .
Paper Structure (9 sections, 10 theorems, 63 equations, 2 figures)

This paper contains 9 sections, 10 theorems, 63 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Preliminaries
  4. Proof of Theorem \ref{['thm:bipartite']}
  5. Inclusions
  6. Equality
  7. The action of $\varphi$ on $H_0$
  8. The action of $\varphi$ on $H_1$
  9. Acknowledgments

Key Result

Theorem 1

Let $m$ and $n$ be positive integers with $m \le n$. Then

Figures (2)

  • Figure 1: A graph $G$ and its $2$-token graph $F_2(G)$.
  • Figure 2: An example of $\pi_\varphi$ on sets $Y(0,1),\ldots,Y(0,n)$, where $n=5$ and $\varphi(H_0)=H_0$.

Theorems & Definitions (19)

  • Theorem 1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Corollary 4.4
  • proof
  • Lemma 4.5
  • ...and 9 more