Let $W_n$ denote the wheel graph having $n$-vertices. If $i$ and $j$ are any two vertices of $W_n$, define \[d_{ ij}:= \begin{ cases} 0 & \mbox{ if}~i=j \\ 1 & \mbox{ if}~i~ \mbox{ and} ~j~ \mbox{ are adjacent} \\ 2 & \mbox{ else}. \end{ cases}\] Let $D$ be the $n \times n$ matrix with $(i,j)^{ \rm th}$ entry equal to $d_{ ij}$. The matrix $D$ is called the distance matrix of $W_n$. Suppose $n \geq 5$ 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 $\widetilde{ L}$ and a rank one matrix $ww'$ such that \[D^\dagger = -\frac{ 1}{ 2} \widetilde{ L}+\frac{ 4}{ n-1}ww'.\] Here, $\widetilde{ L}$ is positive semidefinite, ${ \rm rank}(\widetilde{ L})=n-2$ and all row sums are equal to zero.
Categories: math.CO math.FA