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

Let n≥4 be an even integer and Wn be the wheel graph with n vertices. The distance dij between any two distinct vertices i and j of Wn is the length of the shortest path connecting i and j. Let D be the n×n symmetric matrix with diagonal entries equal to zero and off-diagonal entries equal to dij. In this paper, we find a positive semidefinite matrix L˜ such that rank(L˜)=n−1, all row sums of L˜ equal to zero, and a rank one matrix wwT such that [Formula presented] An interlacing property between the eigenvalues of D and L˜ is also proved.