On Edge-Distance and Edge-Eccentric Graph of a Graph

Abstract

An elementary circuit (or tie) is a subgraph of a graph and the set of edges in this subgraph is called an elementary tieset. The distance $d(e_{i}, e_{j})$ between two edges in an undirected graph is defined as the minimum number of edges in a tieset containing $e_{i}$ and $e_{j}$. The eccentricity $\varepsilon_{\tau}(e_{i})$ of an edge $e_{i}$ is $\varepsilon_{\tau}(e_{i})=\displaystyle \max_{\substack{e_{j}\in E}}d(e_{i}, e_{j})$. In this paper, we have introduced the edge - self centered and edge - eccentric graph of a graph and have obtained results on these concepts.