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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 [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

Citations:

Zbl 1218.05091
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References:

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