## 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

### Keywords:

Hamiltonian cycle; Hamiltonian path; spectral radius
Full Text:

### References:

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