Signless Laplacian Matrix

The signless Laplacian matrix of a graph is

where is the degree matrix and is the adjacency matrix. It differs from the ordinary Laplacian matrix only in the sign of the off-diagonal adjacency term.

The matrix is a positive semidefinite matrix. The multiplicity of its zero eigenvalue is the number of connected components of that are bipartite graphs.

Cai et al. (2026) proved that every triangle-free graph on vertices satisfies . Here is the least eigenvalue, not the signless Laplacian spectral radius. The bound is sharp, with equality for the Higman-Sims graph and its balanced blow-ups. Cai et al. (2026) report GPT-6.1 Sol assistance with proof development, semidefinite programming, and exact checks, with independent author verification.

References

Cai, D.; Chen, Z.; and Zhang, X.-D. "A Sharp Bound for the Least Signless Laplacian Eigenvalue of Triangle-Free Graphs." 10 Oct 2026. https://arxiv.org/abs/2610.14503.

Cvetković, D.; Rowlinson, P.; and Simić, S. K. "Signless Laplacians of Finite Graphs." Linear Algebra Appl. 423, 155-171, 2007. https://doi.org/10.1016/j.laa.2007.01.009.

Li, W.-J.; Wang, Z.; and Guo, J.-M. "The (Signless) Laplacian Spectral Radius with a Fixed Induced Subgraph of a Graph." Elec. J. Combin. 33, P3.81, 2026. https://doi.org/10.37236/12935.

Signless Laplacian Matrix

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