On distance matrices of wheel graphs with odd number of vertices

Abstract

Let $W_n$ denote the wheel graph having $n$-vertices. If $i$ and $j$ are any two vertices of $W_n$, define \[d_{ij}:= \begin{cases} 0 & \mbox{if}~i=j \\ 1 & \mbox{if}~i~ \mbox{and} ~j~ \mbox{are adjacent} \\ 2 & \mbox{else}. \end{cases}\] Let $D$ be the $n\times n$ matrix with $(i,j{)}^{\mathrm{th}}$ entry equal to $d_{ij}$. The matrix $D$ is called the distance matrix of $W_n$. Suppose $n\ge 5$ is an odd integer. In this paper, we deduce a formula to compute the Moore-Penrose inverse of $D$. More precisely, we obtain an $n\times n$ matrix $\tilde{L}$ and a rank one matrix $w{w}^{\prime }$ such that \[D^\dagger = -\frac{1}{2} \widetilde{L}+\frac{4}{n-1}ww'.\] Here, $\tilde{L}$ is positive semidefinite, $\mathrm{rank}(\tilde{L})=n-2$ and all row sums are equal to zero.

References in corpus (1)

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