Inverse of the Squared Distance Matrix of a Complete Multipartite Graph

Let $G$ be a connected graph on $n$ vertices and $d_{ij}$ be the length of the shortest path between vertices $i$ and $j$ in $G$. We set $d_{ii}=0$ for every vertex $i$ in $G$. The squared distance matrix $\mathrm{\Delta }(G)$ of $G$ is the $n\times n$ matrix with $(i,j{)}^{th}$ entry equal to $0$ if $i=j$ and equal to $d_{ij}^2$ if $i\mathrm{\ne }j$. For a given complete $t$-partite graph $K_{n_1,n_2,\cdots ,n_t}$ on $n={\sum }_{i=1}^t{n}_i$ vertices, under some condition we find the inverse $\mathrm{\Delta }(K_{n_1,n_2,\cdots ,n_t}{)}^{-1}$ as a rank-one perturbation of a symmetric Laplacian-like matrix $\mathcal{L}$ with $\text{rank}(\mathcal{L})=n-1$. We also investigate the inertia of $\mathcal{L}$.