Token Graphs

Let Γ be a graph with diameter d ≥ 2. Recall Γ is 1-homogeneous (in the sense of Nomura) whenever for every edge xy of Γ the distance partition

{{z ∈ V(Γ) | ∂(z, y) = i, ∂(x, z) = j} | 0 ≤ i, j ≤ d}

is equitable and its parameters do not depend on the edge ...

Let &Ggr; be a distance-regular graph with diameter at least three and height h = 2, where h = {\rm{max}}\{i : p_{di}^d \ne 0 \} . Suppose that for every ý in &Ggr; and ß in &Ggr; _d (ý), the induced subgraph on &Ggr; _d (ý) ý &Ggr;₂(ß) is a clique. ...

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 are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. It was proved that the algebraic connectivity ...${}_{}$${}_{}$