AI 中文总结
研究邻接度代数与图的谱确定问题，通过证明相关定理，表明标量矩能确定树，对于一般图这些矩是度装饰毛虫同态计数，所得矩刚性类在颜色细化层次结构内且有首次小阶失败情况。
AI 中文摘要
麦凯证明了邻接矩阵\(A\)和对角度矩阵\(D\)的所有多项式函数的谱能确定一棵树。我们证明了该定理的一个主要版本。设\(\mathcal A(G)=\langle I,A_G,D_G\rangle\)且\(M_G=\mathcal A(G)\mathbf1\)为由全\(1\)向量生成的循环模。对于连通图，理想\(\mathcal A(G)J\mathcal A(G)\)（其中\(J=\mathbf1\mathbf1^T\)）作用于\(M_G\)如同全自同态代数。我们表明每个森林满足\(M_G = U_G\)（自同构轨道模），且树的轨道商上的诱导代数是全矩阵代数。由此标量矩\(\mathbf1^Tw(A_T,D_T)\mathbf1\)能确定每棵树。对于一般图，这些矩是度装饰毛虫同态计数。所得的矩刚性类位于可和、紧致、可细化的颜色细化层次结构内，其首次小阶失败是\(M_G\)不可见的十顶点整数切换。
英文摘要
McKay proved that the spectra of all polynomial functions of the adjacency matrix $A$ and the diagonal degree matrix $D$ determine a tree. We prove a principal version of this theorem. Let $\mathcal A(G)=\langle I,A_G,D_G\rangle$ and let $M_G=\mathcal A(G)\mathbf1$ be the cyclic module generated by the all-ones vector. For connected graphs the ideal $\mathcal A(G)J\mathcal A(G)$, where $J=\mathbf1\mathbf1^T$, acts on $M_G$ as the full endomorphism algebra. We show that every forest satisfies $M_G=U_G$, the automorphism-orbit module, and that the induced algebra on the orbit quotient of a tree is a full matrix algebra. It follows that the scalar moments $\mathbf1^Tw(A_T,D_T)\mathbf1$ determine every tree. For general graphs these moments are degree-decorated caterpillar homomorphism counts. The resulting moment-rigidity class lies inside the amenable, compact, refinable hierarchy of color refinement, and its first small-order failures are ten-vertex integral switchings invisible to $M_G$.
