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

TL;DR

This paper introduces a representation-theoretic refinement of the graph Laplacian spectrum focusing on the (n-2,2) component, providing explicit formulas, moments, and applications to tree isomorphism.

Contribution

It explicitly models the (n-2,2) spectral component, derives formulas for moments, and connects these to tree reconstruction and the graph isomorphism problem.

Findings

The (n-2,2) component yields the first correction to the Laplacian spectrum.

Explicit formulas for the first three moments involve subgraph counts.

Weighted trace polynomials can reconstruct trees from the second moment.

Abstract

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 \[ X_G=\sum_{ij\in E(G)} (ij)\in \C[S_n]. \] 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 $m=|E(G)|$. 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 Thi\'ery \cite{Thiery} and formulate a more explicit support-forest-profile conjecture for the unweighted graph isomorphism problem for trees.