AI 中文总结
反驳Akbari等人关于连通图线图符号差至多为1的推测，给出14顶点仙人掌图和48顶点无三角形图两个反例，且由14顶点图副本链接得到的连通图线图符号差无界，无法进行常数界修正。
AI 中文摘要
Akbari、Elphick、Kumar、Pragada和Tang[《离散数学》349(2026)114953]推测，对于每个连通图G，G的线图的正邻接特征值比负邻接特征值最多多一个；等价地，连通线图的符号差至多为1。我们用两个独立找到的反例反驳了该推测：一个由两个五边形通过桥连接到正方形相邻顶点组成的14顶点仙人掌图，通过精确的特征多项式证明其线图的惯性为(9,0,7)，以及一个通过模拟退火找到并经精确有理算术验证的48顶点无三角形图。实际上，将14顶点图的副本链接起来可得到14k个顶点的连通图，其线图的符号差对于每个k>=1都为k+1。因此，连通线图的符号差是无界的，并且该推测不可能有常数界修正。
英文摘要
Akbari, Elphick, Kumar, Pragada and Tang [Discrete Math. 349 (2026) 114953] conjectured that for every connected graph G, the line graph of G has at most one more positive than negative adjacency eigenvalue; equivalently, the signature of a connected line graph is at most 1. We refute the conjecture with two independently found counterexamples: a 14-vertex cactus consisting of two pentagons attached by bridges to adjacent vertices of a square, whose line graph has inertia (9,0,7) by an exact characteristic-polynomial certificate, and a 48-vertex triangle-free graph found by simulated annealing and verified in exact rational arithmetic. Indeed, chaining copies of the 14-vertex graph yields connected graphs on 14k vertices whose line graphs have signature k+1 for every k >= 1. The signature of connected line graphs is therefore unbounded, and no constant-bound repair of the conjecture is possible.
Commentsv2: corrected description of the 48-vertex mechanism; added references to independent work of A. Paone
