On distance matrices of wheel graphs with an 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) := {0 if i = j 1 if i and j are adjacent 2 else. Let D be the n x n matrix with (i, j)th entry equal to d(ij). The matrix D is called the distance matrix of W-n. Suppose n >= 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 x n matrix (L) over tilde and a rank one matrix ww' such that D-dagger = -1/2 (L) over tilde + 4/n - 1ww'. Here, (L) over tilde is positive semidefinite, rank((L) over tilde) = n - 2 and all row sums are equal to zero.