A structural criterion for spectral determination

Identify a structural or algebraic criterion for a graph matrix M that determines whether the spectrum of M distinguishes threshold graphs up to isomorphism, without requiring a separate case-by-case proof for each matrix.

Background

The adjacency, Laplacian, and distance-Laplacian spectra determine connected threshold graphs, whereas the distance, distance-signless-Laplacian, and Seidel spectra do not. The paper seeks a unifying explanation for this division.

A possible criterion could involve an algebraic relationship between M and J, or the twin-class and equitable-quotient structure intrinsic to threshold graphs. The question is also linked to the proposed master-identity criterion for simultaneous integrality and spectral determination.

References

A unifying explanation for why some matrices determine threshold graphs and others do not — beyond case-by-case proof — does not appear to be known, and we record it as Open Problem~\ref{op:unify} below.

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

A graph matrix $M(G)$ is both always-integral and spectrally determining for threshold graphs if and only if $M(G)$ is an affine function of $L(G)$ and $J$ alone (i.e. $M(G) = \alpha L(G) + \beta J + \gamma I$ for scalars $\alpha,\beta,\gamma$, as is the case for $L$ itself and, via Proposition~\ref{prop:master}, for $DL$).

— The Spectral Theory of Threshold Graphs: A Survey  (2610.22838 - Manna et al., 16 Oct 2026) in Conjecture, Section 9.4