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Spectral radius and Hamiltonicity of graphs. (English) Zbl 1218.05091

Summary: Let \(G\) be a graph of order \(n\) and \(\mu (G)\) be the largest eigenvalue of its adjacency matrix. Let \(\overline G\) be the complement of \(G\).
Write \(K_{n-1}+v\) for the complete graph on \(n-1\) vertices together with an isolated vertex, and \(K_{n-1}+e\) for the complete graph on \(n-1\) vertices with a pendent edge.
We show that:
{}If \(\mu (G)\geqslant n-2\), then \(G\) contains a Hamiltonian path unless \(G=K_{n-1}+v\); if strict inequality holds, then \(G\) contains a Hamiltonian cycle unless \(G=K_{n-1}+e\).
{}If \(\mu(\overline G) \leqslant \sqrt{n-1}\), then \(G\) contains a Hamiltonian path unless \(G=K_{n-1}+v\).
{}If \(\mu(\overline G) \leqslant \sqrt{n-2}\), then \(G\) contains a Hamiltonian cycle unless \(G=K_{n-1}+e\).

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C45 Eulerian and Hamiltonian graphs
05C35 Extremal problems in graph theory
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