Inverse of the squared distance matrix of a complete multipartite graph

Let G be a connected graph on n vertices and dᵢⱼ be the length of the shortest path between vertices i and j in G. We set dᵢᵢ=0 for every vertex i in G. The squared distance matrix Δ(G) of G is the n× n matrix with (i,j)ᵗʰ entry equal to $0$ if $i = j$ and equal to dᵢⱼ² if i ≠ j. For a given complete t-partite graph Kn₁,n₂,⋯,nₜ on n=∑ᵢ₌₁ᵗ nᵢ vertices, under some condition we find the inverse Δ(Kn₁,n₂,⋯,nₜ)⁻¹ as a rank-one perturbation of a symmetric Laplacian-like matrix L with rank (L)=n-1. We also investigate the inertia of L.