On distance matrices of helm graphs obtained from wheel graphs with an even number of vertices

Let n≥4. The helm graph Hn on 2n−1 vertices is obtained from the wheel graph Wn by adjoining a pendant edge to each vertex of the outer cycle of Wn. Suppose n is even. Let D:=[dij] be the distance matrix of Hn. In this paper, we first show that det(D)=3(n−1)2n−1. Next, we find a matrix L and a vector u such that D−1=−[Formula presented]L+[Formula presented]uu′. We also prove an interlacing property between the eigenvalues of L and D.