On Eccentricity Matrices of Wheel Graphs

The eccentricity matrix $E(G)$ of a simple connected graph $G$ is obtained from the distance matrix $D(G)$ of $G$ by retaining the largest distance in each row and column, and by defining the remaining entries to be zero. This paper focuses on the eccentricity matrix $E(W_n)$ of the wheel graph $W_n$ with $n$ vertices. By establishing a formula for the determinant of $E(W_n)$, we show that $E(W_n)$ is invertible if and only if $n \not\equiv 1\Mod3$. We derive a formula for the inverse of $E(W_n)$ by finding a vector $\mathbf{w}\in \mathbb{R}^n$ and an $n \times n$ symmetric Laplacian-like matrix $\widetilde{L}$ of rank $n-1$ such that \begin{eqnarray*} E(W_n)^{-1} = -\frac{1}{2}\widetilde{L} + \frac{6}{n-1}\mathbf{w}\mathbf{w^{\prime}}. \end{eqnarray*} Further, we prove an analogous result for the Moore-Penrose inverse of $E(W_n)$ for the singular case. We also determine the inertia of $E(W_n)$.