Spectra of symmetric powers of graphs and the Weisfeiler-Lehman refinements

TL;DR

This paper investigates whether the spectra of symmetric powers of graphs can serve as complete invariants for graph isomorphism, revealing limitations of spectral methods in distinguishing non-isomorphic graphs.

Contribution

It proves that if the Weisfeiler-Lehman method fails to distinguish two graphs, their symmetric powers are cospectral, showing spectral invariants are not complete for graph isomorphism.

Findings

Spectra of symmetric powers are not complete invariants for graph isomorphism.

Failure of the Weisfeiler-Lehman method implies cospectral symmetric powers.

Existence of non-isomorphic graphs with identical symmetric power spectra.

Abstract

The k-th power of a n-vertex graph X is the iterated cartesian product of X with itself. The k-th symmetric power of X is the quotient graph of certain subgraph of its k-th power by the natural action of the symmetric group. It is natural to ask if the spectrum of the k-th power --or the spectrum of the k-th symmetric power-- is a complete graph invariant for small values of k, for example, for k=O(1) or k=O(log n). In this paper, we answer this question in the negative: we prove that if the well known 2k-dimensional Weisfeiler-Lehman method fails to distinguish two given graphs, then their k-th powers --and their k-th symmetric powers-- are cospectral. As it is well known, there are pairs of non-isomorphic n-vertex graphs which are not distinguished by the k-dim WL method, even for k=Omega(n). In particular, this shows that for each k, there are pairs of non-isomorphic n-vertex graphs with cospectral k-th (symmetric) powers.