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The line graph of a graph $G$, denoted by $L\left(G\right)$, has $E\left(G\right)$ as its vertex set, where two vertices in $L\left(G\right)$ are adjacent if and only if the corresponding edges in $G$ have a vertex in common. For a graph $H$, define ${\overline{\sigma }}_2\left(H\right)=min\{d\left(u\right)+d\left(v\right):uv\in E\left(H\right)\}$. Let $H$ be a 2-connected claw-free simple graph of order $n$ with $\delta \left(H\right)\ge 3$. We show that, if ${\overline{\sigma }}_2\left(H\right)\ge \frac{1}{7}(2n-5)$ and $n$ is sufficiently large, then either $H$ is traceable or the Ryjáček’s closure $\mathrm{cl}\left(H\right)=L\left(G\right)$, where $G$ is an essentially $2$-edge-connected triangle-free graph that can be contracted to one of the two graphs of order 10 which have...