Inverse of the squared distance matrix of a complete multipartite graph

Abstract

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 $\Delta(G)$ 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 $i \neq j$. 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 $\Delta(K_{n_1,n_2,\cdots,n_t})^{-1}$ 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 $\mathcal{L}$.