发表机构
College of Mathematics and System Sciences, Xinjiang University; Department of Mathematics, Millersville University; School of Mathematics and Systems Science, Guangdong Polytechnic Normal University; Department of Mathematics, West Virginia University（新疆大学数学与系统科学学院; 米勒维尔大学数学系; 广东技术师范大学数学与系统科学学院; 西弗吉尼亚大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究确定了二阶迭代线图的最优连通性函数κ_{L²}(d,k)的精确表达式，完善了Niepel与Knor的相关结论。
AI 中文摘要
图G的线图L(G)定义为简单图，其顶点是G的边，L(G)中两顶点相邻当且仅当G中对应边有公共顶点，记L²(G)=L(L(G))。对于正整数d和k，函数κ_{L²}(d,k)=inf{κ(L²(G)): κ'(G)≥k且δ(G)≥d}已被研究。Niepel和Knor证明，对任意整数d≥3，κ_{L²}(d,1)≥d-1。本研究证明：若d≥3且k≥1，则κ_{L²}(d,k)=min{f(d,k),4d-6}，其中f(d,k)分段定义：当1≤k≤⌊d/2⌋时，f(d,k)=k(d−k)；当⌊d/2⌋<k<(3d−1)/4时，f(d,k)=kd−k²+2k⌈d/2⌉−⌈d/2⌉·d；当(3d−1)/4≤k<d时，f(d,k)=kd−k²+2k⌊d/2⌋−2⌊d/2⌋²；当d=k时，f(d,k)=d⌈d/2⌉。
英文摘要
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 $\frac{3d-1}{4}\leq k<d$, }\\ d(\lceil \frac{d}{2}\rceil), & \mbox{ if } d=k. \end{array} \right.\nonumber \end{equation}
Commentsv2: Revised proofs and corrected minor typos
