On the connectivity and diameter of Token graphs from a vertex induced sub-graph perspective

TL;DR

This paper investigates the connectivity and diameter of token graphs derived from vertex-induced sub-graphs, revealing their properties and relationships with underlying graphs, especially when the base graph has diameter 2.

Contribution

It provides new insights into the vertex connectivity and diameter of token graphs, linking these properties to the structure of the original graph and exploring combinatorial relationships.

Findings

Vertex connectivity equals minimal degree in token graphs.

Diameter of token graphs is characterized when the base graph has diameter 2.

New combinatorial relationships between $L$ and its token graph are established.

Abstract

Token graphs, or symmetric powers of graphs, see \cite{alavi2002survey} and \cite{Fabila-Monroy2012}, are defined on the $k$-combinations of the vertex set of some graph $L$, where edges exist between two such combinations, if their symmetric difference corresponds to an edge in the underlying graph $L$. It has been noted, for example in \cite{AUDENAERT200774}, that these graphs constitute an inherent correspondence between the relationships between random walks and graph invariants, and particle systems and higher order graph properties, employing in particular the structure of vertex induced sub-graphs. In this work, we contribute to this perspective, by giving a synthetic perspective on the vertex connectivity of token graphs, which equals its minimal degree, as well as on their diameter, if the underlying graph $L$ has diameter $2$. Some combinatorial results on the clique-Johnson graph link between $L$ and its token graph are proven as well.