On distance matrices of wheel graphs with odd number of vertices

Let $W_n$ denote the wheel graph having $n$-vertices. If $i$ and $j$ are any two vertices of $W_n$, define $$d_{ij}:=\{\begin{array}{ll}0& ifi=j\\ 1& ifiandjareadjacent\\ 2& else\mathrm{.}\end{array}$$ Let $D$ be the $n\times n$ matrix with $(i,j{)}^{\mathrm{t}\mathrm{h}}$ entry equal to $d_{ij}$. The matrix $D$ is called the distance matrix of $W_n$. Suppose $n\ge 5$ is an odd integer. In this paper, we deduce a formula to compute the Moore-Penrose inverse of $D$. More precisely, we obtain an $n\times n$matrix$\widetilde{L}$andarankonematrix$ww’$ such that $$D^{†}=-\frac{1}{2}\tilde{L}+\frac{4}{n-1}w{w}^{\mathrm{\prime }}\mathrm{.}$$ Here, $\tilde{L}$ is positive semidefinite, $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(\tilde{L})=n-2$ and all row sums are equal to zero.