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

Abstract

Let $n \geq 4$. The helm graph $H_n$ on $2n-1$ vertices is obtained from the wheel graph $W_n$ by adjoining a pendant edge to each vertex of the outer cycle of $W_n$. Suppose $n$ is even. Let $D := [d_{ij}]$ be the distance matrix of $H_n$. In this paper, we first show that $\det(D) = 3(n-1)2^{n-1}.$ Next, we find a matrix $\L$ and a vector $u$ such that \[D^{-1} = -\frac{1}{2}\L+\frac{4}{3(n-1)}uu'.\] We also prove an interlacing property between the eigenvalues of $\L$ and $D$.