反转带长悬挂路径的Mostar线图不等式
Reversing the Mostar line-graph inequality with long pendant paths
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中文总结 AI 辅助
本文研究悬挂路径长度对线图与原图Mostar指数差的影响，证明其仿射性，并解决Alex-Indulal问题，给出正圈数下最大度为3的无穷多反例及相等性条件。
中文摘要 AI 辅助
设$K_t$是通过在固定有根图$K$上附加一条长度为$t$的悬挂路径得到的图。我们证明$\mathrm{Mo}(L(K_t))-\mathrm{Mo}(K_t)$，即其线图与原图Mostar指数之差，在尖锐均匀截断之后，在每个奇偶类上恰好是仿射的。斜率取决于距离层邻居计数，并且对每个非二分核心都是正的。三角形链给出了无穷多个图，使得$\mathrm{Mo}(L(G))>\mathrm{Mo}(G)$，且在每个正圈数$c$下最大度为3，解决了Alex-Indulal问题3.3。在零斜率时，仙人掌分支质量公式决定相等性；相同的顶点剖面不一定给出相同的截距。当且仅当$c$为奇数时，存在固定的核心使得对所有足够长的附加都相等。最大度为3就足够了，并且在每个偶数$c$下，二叉树构造在一个最终的奇偶类上给出相等性。
英文摘要
Let $K_t$ be obtained by attaching a pendant path of length $t$ to a fixed rooted graph $K$. We prove that $\mathrm{Mo}(L(K_t))-\mathrm{Mo}(K_t)$, the difference between its line-graph and original Mostar indices, is exactly affine on each parity class beyond a sharp uniform cutoff. The slope depends on distance-level neighbor counts and is positive for every non-bipartite core. Triangle chains give infinitely many graphs with $\mathrm{Mo}(L(G))>\mathrm{Mo}(G)$ at every positive cyclomatic number $c$, with maximum degree three, resolving Alex-Indulal Problem 3.3. At zero slope, a cactus branch-mass formula decides equality; identical vertex profiles need not give identical intercepts. Fixed cores giving equality for all sufficiently long attachments exist exactly when $c$ is odd. Maximum degree three suffices, and at every even $c$ a binary-tree construction gives equality on one eventual parity class.
