2026/07/23 by Zhipeng Lu, Pengxiang Li
#math.CO #math.SP
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)=⟨ I,AG,DG⟩ and let MG=\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\mathbf1T, acts on MG as the full endomorphism algebra. We show that every forest satisfies MG=UG, 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 \mathbf1Tw(AT,DT)\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 MG.