An inverse formula for the distance matrix of a fan graph

Let F-n be the fan graph with n >= 3 vertices. The distance d(i,j) between any two distinct vertices i and j of F-n is the length of the shortest path connecting land j. Let (D) over cap be the n x n symmetric matrix with diagonal entries equal to zero and off-diagonal entries equal to d(i,j). In this paper, we find two positive semidefinite matrices L-o and L-e such that rank(L-o) = rank(L-e) = n - 1, all row sums of L-o and L-e, are equal to zero, and find two rank one matrices alpha alpha' and (alpha) over tilde(alpha) over tilde' such that (D) over cap (-1) = {-1/n L-o + 4/n(n(2)-1)alpha alpha', if n is odd; -1/nL(e) + 1/n (alpha) over tilde(alpha) over tilde', if n is even. The interlacing property between the eigenvalues of (D) over cap and L-o (resp., L-e ) is also established.