## 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
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### References:

 [1] Bondy, A.; Chvatal, V., A method in graph theory, Discrete math., 15, 111-135, (1976) · Zbl 0331.05138 [2] S. Butler, F. Chung, Small spectral gap in the combinatorial Laplacian implies Hamiltonian, Ann. Combin., in press. · Zbl 1229.05193 [3] van den Heuvel, J., Hamilton cycles and eigenvalues of graphs, Linear algebra appl., 226-228, 723-730, (1995) · Zbl 0846.05059 [4] Hofmeister, M., Spectral radius and degree sequence, Math. nachr., 139, 37-44, (1988) · Zbl 0695.05046 [5] Krivelevich, M.; Sudakov, B., Sparse pseudo-random graphs are Hamiltonian, J. graph theory, 42, 17-33, (2003) · Zbl 1028.05059 [6] Mohar, B., A domain monotonicity theorem for graphs and Hamiltonicity, Discrete appl. math., 36, 169-177, (1992) · Zbl 0765.05071 [7] Ore, O., Note on Hamilton circuits, Amer. math. monthly, 67, 55, (1960) · Zbl 0089.39505 [8] Stanley, R., A bound on the spectral radius of graphs with $$e$$ edges, Linear algebra appl., 87, 267-269, (1987) · Zbl 0617.05045
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