## Signless Laplacian spectral radius and Hamiltonicity.(English)Zbl 1188.05086

For an $$n$$ vertex of a graph $$G$$, the matrix $$L^*(G)=D(G)+A(G)$$ is the signless Laplacian matrix of $$G$$, where $$D(G)$$ is the diagonal matrix of vertex degrees and $$A(G)$$ is the adjacency matrix of $$G$$. Let $$\gamma(G)$$ be the largest eigenvalue of $$L^*(G)$$. The author shows that if $$\gamma(\bar G)\leq n$$ then $$G$$ contains a Hamiltonian path and if $$\gamma(\bar G)\leq n-1$$ then $$G$$ contains a Hamiltonian cycle, except in a few fully characterized cases. This work uses the techniques and approach from [M. Fiedler and V. Nikiforov, “Spectral radius and Hamiltonicity of graphs”, Linear Algebra Appl. 432, 2170–2173 (2010; Zbl 1218.05091)].

### MSC:

 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C35 Extremal problems in graph theory

Zbl 1218.05091
Full Text:

### References:

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