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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 of the usual Hamiltonian graphs. In this paper, we present some spectral sufficient conditions for a graph to be k-edge-Hamiltonian and k-Hamiltonian in terms of the adjacency spectral radius as well as the signless Laplacian spectral radius. Our results could be viewed as slight extensions of the recent theorems proved by Li and Ning [Linear Multilinear Algebra. 2016;64:2252–2269], Nikiforov [Czechoslovak Math J. 2016;66:925–940] and Li et al. [Linear Multilinear Algebra. 2018;66:2011–2023]. Moreover, we shall prove a stability result for graphs being k-Hamiltonian, which could be regarded as a complement of two recent results of Füredi et al. [Discrete Math. 2017;340:2688–2690] and [Discrete Math. 2019;342:1919–1923]. [ABSTRACT FROM AUTHOR]

We present an Oppenheim type determinantal inequality for positive definite block matrices. Recently, Lin [Linear Algebra Appl. 2014;452:1–6] proved a remarkable extension of Oppenheim type inequality for block matrices, which solved a conjecture of Günther and Klotz. There is a requirement that two matrices commute in Lin's result. The motivation of this paper is to obtain another natural and general extension of Oppenheim type inequality for block matrices to get rid of the requirement that two matrices commute. [ABSTRACT FROM AUTHOR]