## Small spectral gap in the combinatorial Laplacian implies Hamiltonian.(English)Zbl 1229.05193

Summary: We consider the spectral and algorithmic aspects of the problem of finding a Hamiltonian cycle in a graph. We show that a sufficient condition for a graph being Hamiltonian is that the nontrivial eigenvalues of the combinatorial Laplacian are sufficiently close to the average degree of the graph. An algorithm is given for the problem of finding a Hamiltonian cycle in graphs with bounded spectral gaps which has complexity of order $$n^{c\ln n}$$.

### MSC:

 05C45 Eulerian and Hamiltonian graphs 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C85 Graph algorithms (graph-theoretic aspects)

### Keywords:

Hamiltonian; combinatorial Laplacian; spectral graph theory
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### References:

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