Edge-Neighbor-Rupture Degree of Graphs

Abstract

The edge-neighbor-rupture degree of a connected graph $G$ is defined to be $\text{ENR}\left(G\right)=\text{max}\left\{\omega \left(G-S\right)-\left|S\right|-m\left(G-S\right):S\subseteq E\left(G\right),\omega \left(G-S\right)\ge 1\right\}$, where $S$ is any edge-cut-strategy of $G$, $\omega \left(G-S\right)$ is the number of the components of $G-S$, and $m\left(G-S\right)$ is the maximum order of the components of $G-S$. In this paper, the edge-neighbor-rupture degree of some graphs is obtained and the relations between edge-neighbor-rupture degree and other parameters are determined.