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 semi-definite 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)^{\dagger} = -\frac{1}{2}L +\frac{4}{3(n-1)}\mathbf{w}\mathbf{w^{\prime}},\end{eqnarray*}where the rank of $L$ is $2n-3$. We also investigate the inertia of $D(H_n)$.