To determine the signs of the eigenvalues of a graph's adjacency matrix given their magnitudes, we can leverage properties of the matrix's characteristic polynomial and trace constraints. Here's the key analysis:

Approach Using Trace Constraints

The adjacency matrix ( A ) of a (non-bipartite) graph is real symmetric, so its eigenvalues ( \lambda_1, \ldots, \lambda_n ) are real. Given thei...

Yes. You can recover the signs without computing the eigenvalues themselves, by using traces of odd powers (or, equivalently, inertia/Sturm counts). Here are two standard ways.

1. Using odd power traces and a Vandermonde system

• Let the distinct absolute values be b1 < b2 < … < br, with total multiplicities mj (so Σj mj = n). For each j, let pj be the multiplicity of +bj and nj the multiplicity...

Computing Eigenvalue Signs Given Their Magnitudes

This is an interesting question at the intersection of spectral graph theory and computational complexity.

When we know only the magnitudes of eigenvalues |λ₁|, |λ₂|, ..., |λₙ| of an adjacency matrix, determining their signs efficiently is not straightforward. Here's what we can say:

What We Know

1. For an adjacency matrix of an undirected...

Let G be a simple, undirected, non–bipartite graph and let

 A ∈ ℤ^{n×n}

be its (symmetric) adjacency matrix.
Assume we are given the multiset

 {|λ₁| , … , |λ_n|} , λ₁ ≥ … ≥ λ_n

and we want to decide, for every i, whether λ_i is +|λ_i| or –|λ_i|.

1. What information is still missing?
   We have lost exactly one bit of information per eigenvalue—the sign.
   Any algorithm must th...

Below is a summary of why, as far as we know, there is no “magic shortcut’’ that, given the list of absolute‐values {|λ₁|,…,|λₙ|}, recovers the signs sign(λᵢ) in o(n³) (or o(n^ω)) time without essentially re‐doing a spectral computation.

1. The problem really is to decide, for each magnitude aᵢ, whether the corresponding eigenvalue of A is +aᵢ or −aᵢ. Equivalently one needs an eigenvector vᵢ fo...

This is a fascinating question that touches on the relationship between the combinatorial structure of a graph and the algebraic properties of its adjacency matrix. The short answer is that there is likely no general method that is significantly faster than re-computing the eigenvalues, because determining the signs from the magnitudes is an NP-hard problem.

Let's break down why this is the c...

Let

 A ∈ {0,1}n×n

be the symmetric adjacency matrix of an undirected (and not necessarily bipartite) graph, and let

 |λ1| ≥ |λ2| ≥ … ≥ |λn|

be the multiset of absolute values of its eigenvalues that we are given.
We want to recover the signs εi ∈ {–1 , +1} so that λi = εi |λi| for every i.

1. What information can be extracted trivially?

• In O(n) time we can find out whether t...

<think> We are given the absolute values of the eigenvalues of the adjacency matrix A of a non-bipartite graph. We wish to compute the signs of these eigenvalues without having to recompute the eigenvalues themselves.

Recall that for an adjacency matrix of a graph:

• The trace of A is zero (since there are no self-loops, the diagonal is zero). Therefore, the sum of the eigenvalues is zero: λ...

This is an interesting question about the computational complexity of determining eigenvalue signs given their absolute values.

Direct Answer

Unfortunately, there is no known method that is significantly faster than recomputing the eigenvalues themselves in the general case. Here's why:

Key Observations

1. Information-Theoretic Perspective: The signs of eigenvalues represent bits...