The Moore-Penrose Inverse of the Distance Matrix of a Helm Graph

In this paper, we give necessary and sufficient conditions for a real symmetric matrix, and in particular, for the distance matrix $D(H_n)$ of a helm graph $H_n$ to have their Moore-Penrose inverses as the sum of a symmetric Laplacian-like matrix and a rank one matrix. As a consequence, we present a short proof of the inverse formula, given by Goel (Linear Algebra Appl. 621:86–104, 2021), for $D(H_n)$ when $n$ is even. Further, we derive a formula for the Moore-Penrose inverse of singular $D(H_n)$ that is analogous to the formula for $D(H_n{)}^{-1}$. Precisely, if $n$ is odd, we find a symmetric positive semidefinite Laplacian-like matrix $L$ of order $2n-1$ and a vector $\mathbf{w}\in {\mathbb{R}}^{2n-1}$ such that \begin{eqnarray*} D(H_n)\ssymbol{2} = -\frac{1}{2}L + \frac{4}{3(n-1)}$\mathbf{w}$${\mathbf{w}}^{\mathbf{\prime }}$, \end{eqnarray*} where the rank of $L$ is $2n-3$. We also investigate the inertia of $D(H_n)$.