The (n-2,2)-Spectrum of a Graph

We study a representation-theoretic refinement of the ordinary Laplacian spectrum of a graph. Given a graph $G$ on $n$ vertices, one may associate to it the element $$X_G=\sum _{ij\in E(G)}(ij)\in \text{C}[S_n]\mathrm{.}$$ The action of $X_G$ in irreducible representations of $S_n$ produces spectral invariants of graphs. The standard representation $(n-1,1)$ recovers the ordinary graph Laplacian spectrum, up to the elementary affine change $X_G=mI-L_G$, where $m=\mathrm{\mid }E(G)\mathrm{\mid }$. The next component, $(n-2,2)$, gives the first representation-theoretic correction. We give an explicit edge-space model for this component, derive a concrete coordinate formula for the induced operator, give a conceptual formula for all trace moments, specialize it to trees as universal linear combinations of support-forest counts, and then compute the first three moments explicitly. The third moment is expressed in terms of three-edge subgraph counts. We also introduce a weighted trace polynomial and prove that this weighted refinement already reconstructs every tree from the second moment, except for a single exceptional value of $n$ where the fourth moment suffices. Finally we discuss the relation with the invariant-theoretic approach of Thiéry \cite{Thiery} and formulate a more explicit support-forest-profile conjecture for the unweighted graph isomorphism problem for trees.