An inverse formula for the distance matrix of a wheel graph with even number of vertices

Let $n\ge 4$ be an even integer and $W_n$ be the wheel graph with $n$ vertices. The distance $d_{ij}$ between any two distinct vertices $i$ and $j$ of $W_n$ is the length of the shortest path connecting $i$ and $j$. Let $D$ be the $n\times n$ symmetric matrix with diagonal entries equal to zero and off-diagonal entries equal to $d_{ij}$. In this paper, we find a positive semidefinite matrix $\tilde{L}$ such that $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(\tilde{L})=n-1$, all row sums of $\tilde{L}$ equal to zero and a rank one matrix $w{w}^T$ such that $$D^{-1}=-\frac{1}{2}\tilde{L}+\frac{4}{n-1}w{w}^T\mathrm{.}$$ An interlacing property between the eigenvalues of $D$ and $\tilde{L}$ is also proved.