On the eccentricity inertia indices of chain graphs

Abstract

For a given graph G, the eccentricity matrix of it, written as $\epsilon \left(G\right)$, is created by retaining the largest non-zero entries for each row and column of the distance matrix, while filling the rest with zeros, i.e.,$\epsilon {\left(G\right)}_{uv}=\{\begin{array}{cc}d(u,v),& \text{if}d(u,v)=\min \left\{\epsilon \right(u),\epsilon (v\left)\right\},\\ 0,& \text{otherwise},\end{array}$ where $\epsilon \left(u\right)$ denotes the eccentricity of a vertex u. The eccentricity inertia index of a graph G is represented by a triple $\left(n_{+}\right(\epsilon \left(G\right))$, $n_0\left(\epsilon \right(G\left)\right)$, $n_{-}\left(\epsilon \right(G\left)\right))$, where $n_{+}\left(\epsilon \right(G\left)\right)$ (resp., $n_0\left(\epsilon \right(G\left)\right),n_{-}\left(\epsilon \right(G\left)\right)$) is the count of positive (resp., zero, negative) eigenvalues of $\epsilon \left(G\right)$. In this paper, for each chain graph (a graph which does not contain $C_3,C_5$, or $2K_2$ as induced subgraphs), the eccentricity inertia index of it is completely determined.