Essential edge connectivity of line graphs

Let $G$ be a graph and $L\left(G\right)$ be its line graph. In 1969, Chartrand and Stewart proved that ${\kappa }^{\prime }\left(L\left(G\right)\right)\ge 2{\kappa }^{\prime }\left(G\right)?2$, where ${\kappa }^{\prime }\left(G\right)$ and ${\kappa }^{\prime }\left(L\left(G\right)\right)$ denote the edge connectivity of $G$ and $L\left(G\right)$ respectively. We show a similar relationship holds for the essential edge connectivity of $G$ and $L\left(G\right)$, written ${\kappa }_e^{\prime }\left(G\right)$ and ${\kappa }_e^{\prime }\left(L\left(G\right)\right)$, respectively. In this note, it is proved that if $L\left(G\right)$ is not a complete graph and $G$ does not have a vertex of degree two, then ${\kappa }_e^{\prime }\left(L\left(G\right)\right)\ge 2{\kappa }_e^{\prime }\left(G\right)?2$. An immediate corollary is that $\kappa \left(L^2\left(G\right)\right)\ge 2\kappa \left(L\left(G\right)\right)?2$ for such graphs $G$, where the vertex connectivity of the line graph $L\left(G\right)$ and the second iterated line graph $L^2\left(G\right)$ are written as $\kappa \left(L\left(G\right)\right)$ and $\kappa \left(L^2\left(G\right)\right)$ respectively.