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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 [{\it M. Fiedler} and {\it V. Nikiforov}, “Spectral radius and Hamiltonicity of graphs”, Linear Algebra Appl. 432, 2170--2173 (2010; Zbl 1218.05091)].

05C50Graphs and linear algebra
05C35Extremal problems (graph theory)
Full Text: DOI
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