Main signless Laplacian eigenvalues of quasi-threshold graphs

Characterize connected quasi-threshold graphs with k main signless Laplacian eigenvalues for every integer k≥3, extending the complete characterization currently available for k=2 and the partial results for k=3.

Background

Quasi-threshold graphs, also known as trivially perfect graphs, strictly contain threshold graphs. Structural characterizations of quasi-threshold graphs with prescribed numbers of main signless Laplacian eigenvalues have been developed using condensed quotient matrices.

The case k=2 is resolved, while the case k=3 has only partial results, including results for trees, unicyclic graphs, and bicyclic graphs. A general characterization remains unavailable.

References

The characterization of quasi-threshold graphs with $k$ main signless Laplacian eigenvalues (Section~\ref{sec:signless}) is complete for $k=2$ but only partial for $k=3$ ; a full characterization for general $k$ remains open.

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