On the algebraic connectivity of token graphs

TL;DR

This paper investigates the algebraic connectivity of token graphs, proving a conjecture that relates it to the original graph's connectivity for specific graph families.

Contribution

It proves the conjecture that algebraic connectivity of token graphs equals that of the original graph for new classes like trees and high-degree graphs.

Findings

Proved the conjecture for trees.

Established the conjecture for graphs with large maximum degree.

Extended understanding of spectral properties of token graphs.

Abstract

We study the algebraic connectivity (or second Laplacian eigenvalue) of token graphs, also called symmetric powers of graphs. The $k$-token graph $F_k(G)$ of a graph $G$ is the graph whose vertices are the $k$-subsets of vertices from $G$, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in $G$. Recently, it was conjectured that the algebraic connectivity of $F_k(G)$ equals the algebraic connectivity of $G$. In this paper, we prove the conjecture for new infinite families of graphs, such as trees and graphs with maximum degree large enough.