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)].


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


Zbl 1218.05091
Full Text: DOI


[1] Bondy, A.; Chvatal, V., A method in graph theory, Discrete math., 15, 111-135, (1976) · Zbl 0331.05138
[2] de Caen, D., An upper bound on the sum of squares of degrees in a graph, Discrete math., 185, 245-248, (1998) · Zbl 0955.05059
[3] Cvetković, D.; Doob, M.; Sachs, H., Spectra of graphs – theory and application, (1995), Johann Ambrosius Barth Heidelberg · Zbl 0824.05046
[4] Cvetković, D.; Rowlinson, P.; Simić, S., Signless Laplacians of finite graphs, Linear algebra appl., 423, 155-171, (2007) · Zbl 1113.05061
[5] M. Fiedler, V. Nikiforov, Spectral radius and Hamiltonicity of graphs, Linear Algebra Appl., in press, doi:10.1016/j.laa.2009.01.005. · Zbl 1218.05091
[6] C.St.J.A. Nash-Williams, Valency sequences which force graphs to have Hamiltonian Circuits, University of Waterloo Research Report, Waterloo, Ontario, 1969.
[7] Nikiforov, V., The sum of the squares of degrees: sharp asymptotics, Discrete math., 307, 3187-3193, (2007) · Zbl 1127.05054
[8] Ore, O., Note on Hamilton circuits, Amer. math. monthly, 67, 55, (1960) · Zbl 0089.39505
[9] Peled, U.N.; Pedreschi, R.; Sterbini, Q., \((n, e)\)-graphs with maximum sum of squares of degrees, J. graph theory, 31, 283-295, (1999) · Zbl 0945.05035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.