Distance signless Laplacian spectrum and spectral determination

Determine an explicit closed-form distance signless Laplacian spectrum for threshold graphs from their creation sequences, and determine whether threshold graphs are uniquely determined up to isomorphism by their distance signless Laplacian spectra.

Background

The distance signless Laplacian matrix is defined as DQ(G)=Tr(G)+D(G). Unlike the distance Laplacian, it does not follow from a master identity involving the ordinary Laplacian. The paper derives local twin-class eigenvalues and a quotient-matrix description, but notes that threshold-graph-specific consequences analogous to the distance-Laplacian theory have not been fully developed.

The unresolved issues are to obtain a closed-form spectrum directly from the creation sequence and to settle spectral determination. The latter is known to fail in general because explicit non-isomorphic DQ-cospectral threshold graphs exist, but a complete characterization of DQ-cospectrality is not given.

References

We list this as Open Problem~\ref{op:dqthreshold} in Section~\ref{sec:open}.

— The Spectral Theory of Threshold Graphs: A Survey  (2610.22838 - Manna et al., 16 Oct 2026) in Open Problem, Section 6.3; Open Problem 1, Section 12

Section~\ref{sec:integral} characterizes $A$-, $S$-, $D$-, $DQ$-, and $\varepsilon$-integrality completely for the two-block family $CS(s,t)$, but the general ($m\ge2$) case remains open for every one of these five matrices.

— The Spectral Theory of Threshold Graphs: A Survey  (2610.22838 - Manna et al., 16 Oct 2026) in Section 9.3; Open Problem, Section 12