Sufficient spectral conditions for graphs being $k$-edge-Hamiltonian or $k$-Hamiltonian

A graph G is k-edge-Hamiltonian if any collection of vertex-disjoint paths with at most k edges altogether belong to a Hamiltonian cycle in G. A graph G is k-Hamiltonian if for all S ⊆ V (G) with |S| ≤ k, the subgraph induced by V (G) \ S has a Hamiltonian cycle. These two concepts are classical extensions for the usual Hamiltonian graphs. In this paper, we present some spectral sufficient conditions for a graph to be k-edgeHamiltonian and k-Hamiltonian in terms of the adjacency spectral radius as well as the signless Laplacian spectral radius. Our results extend the recent works proved by Li and Ning [Linear Multilinear Algebra 64 (2016)], Nikiforov [Czechoslovak Math. J. 66 (2016)] and Li, Liu and Peng [Linear Multilinear Algebra 66 (2018)]. Moreover, we shall prove a stability result for graphs being k-Hamiltonian, which can be viewed as a complement of two recent results of Füredi, Kostochka and Luo [Discrete Math. 340 (2017)] and [Discrete Math. 342 (2019)].

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