Eccentricity-spectrum determination

Determine whether connected threshold graphs are uniquely determined up to isomorphism by their eccentricity spectra.

Background

The eccentricity matrix of a threshold graph admits quotient-matrix methods and several eigenvalue-location results, including eigenvalue-free intervals. However, these results do not settle whether two non-isomorphic threshold graphs can share the same eccentricity spectrum.

The spectral-determination status of the eccentricity matrix is consequently listed as open, in contrast with the settled status for adjacency, Laplacian, and distance-Laplacian spectra.

References

Whether threshold graphs are determined by their eccentricity spectrum does not appear to have been settled in the literature surveyed here, despite the detailed eigenvalue-location results of Theorems~\ref{thm:eccthreshold}--\ref{thm:eccentricity2022}.

— The Spectral Theory of Threshold Graphs: A Survey  (2610.22838 - Manna et al., 16 Oct 2026) in Open Problem, Section 12; Table 2, Section 7