On the Connectivity of Token Graphs of Trees

Abstract

Let $k$ and $n$ be integers such that $1\leq k \leq n-1$, and let $G$ be a simple graph of order $n$. The $k$-token graph $F_k(G)$ of $G$ is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their symmetric difference is an edge of $G$. In this paper we show that if $G$ is a tree, then the connectivity of $F_k(G)$ is equal to the minimum degree of $F_k(G)$.

Token graphs of Cayley graphs as lifts

“…. , 11, yield sp(L(H)) = {6 [1] , 3 [6] , 2 [1] , 0 [2] , −2 [12] }. k /n vertices with voltages on the group Z n .…”

“…Some properties of token graphs were studied by Fabila-Monroy, Flores-Peñaloza, Huemer, Hurtado, Urrutia, and Wood [11], including the connectivity, diameter, clique and chromatic number, and Hamiltonian paths. In particular, the connectivity of token graphs of trees was studied by Fabila-Monroy, Leaños, and Trujillo-Negrete in [12].…”