{ "abstract": "The line graph $L(G)$ of a graph $G$ is defined to be the simple graph whose vertices are the edges of $G$, where two vertices in $L(G)$ are adjacent if and only if the corresponding edges in $G$ are incident with a common vertex, and define $L^2(G)=L(L(G))$. For positive integers $d$ and $k$, the function $κ_{L^2}(d,k) = \\inf\\{κ(L^2(G)): κ'(G) \\ge k \\mbox{ and } δ(G) \\ge d\\}$ has been investigated. Niepel and Knor proved that $κ_{L^2}(d,1)\\geq d-1$, for any integer $d \\ge 3$. In this research, it is proved that if $d\\geq 3$ and $k\\geq 1$, then $κ_{L^2}(d,k)= \\min\\{f(d,k), 4d-6\\}$, where \\begin{equation} f(d,k) = \\left\\{ \\begin{array}{ll} k(d-k), & \\mbox{ if $1\\leq k\\leq \\lfloor\\frac{d}{2}\\rfloor$, } \\\\ kd-k^2+2k(\\lceil \\frac{d}{2}\\rceil)-(\\lceil\\frac{d}{2}\\rceil)d, & \\mbox{ if $\\lfloor\\frac{d}{2}\\rfloor< k <\\frac{3d-1}{4}$, } \\\\ kd-k^2+2k\\lfloor \\frac{d}{2}\\rfloor-2(\\lfloor \\frac{d}{2}\\rfloor)^2, & \\mbox{ if $\\frac{3d-1}{4}\\leq k