关于整数支配根猜想
On the Integer Domination Root Conjecture
中文总结 AI 辅助
本文针对整数支配根猜想，构造了阶数为33的反例图G₃₃，证明其支配多项式在x=-4处存在整数根，揭示了转移矩阵与S-单位分支抵消的结构机制。
中文摘要 AI 辅助
支配整数根猜想断言，对于任意图G，支配多项式D(G, x)的仅有的整数根为0和-2。本文给出了一个阶数n=33的反例，其在x=-4处存在整数支配根。我们提供了图G₃₃的完整结构描述，给出了其精确支配多项式D(G₃₃, x)，并证明了其精确有理因式分解。此外，我们概述了涉及转移矩阵和S-单位分支抵消的结构装置机制，该机制导致在x=-4处出现非平凡零点评估。
英文摘要
The domination integer root conjecture asserted that $0$ and $-2$ are the only integer roots of the domination polynomial $D(G, x)$ for any graph $G$. In this paper, we document a counterexample of order $n = 33$ possessing an integer domination root at $x = -4$. We provide the complete structural description of the graph $G_{33}$, present its exact domination polynomial $D(G_{33}, x)$, and demonstrate its exact rational factorization. Furthermore, we outline the structural gadget mechanism involving transfer matrices and $S$-unit branch cancellations that gives rise to non-trivial zero evaluation at $x = -4$.
