The Distance Matrices of Some Graphs Related to Wheel Graphs

Abstract

Let $D$ denote the distance matrix of a connected graph $G$. The inertia of $D$ is the triple of integers ($n_{+}(D), n_0(D), n_{-}(D)$), where $n_{+}(D)$, $n_{\mathrm{0}}(D)$, and $n_{-}(D)$ denote the number of positive, 0, and negative eigenvalues of $D$, respectively. In this paper, we mainly study the inertia of distance matrices of some graphs related to wheel graphs and give a construction for graphs whose distance matrices have exactly one positive eigenvalue.