Hamiltonian s-properties and eigenvalues of k-connected graphs

Abstract

Chvátal and Erdös (1972) [5] proved that, for a k-connected graph G, if the stability number α(G)≤k−s, then G is Hamilton-connected (s=1) or Hamiltonian (s=0) or traceable (s=−1). Motivated by the result, we focus on tight sufficient spectral conditions for k-connected graphs to possess Hamiltonian s-properties. We say that a graph possesses Hamiltonian s-properties, which means that the graph is Hamilton-connected if s=1, Hamiltonian if s=0, and traceable if s=−1.For a real number a≥0, and for a k-connected graph G with order n, degree diagonal matrix D(G) and adjacency matrix A(G), we have identified best possible upper bounds for the spectral radius λ1(aD(Γ)+A(Γ)), where Γ is either G or the complement of G, to warrant that G possesses Hamiltonian s-properties. Sufficient conditions for a graph G to possess Hamiltonian s-properties in terms of upper bounds for the Laplacian spectral radius as well as lower bounds of the algebraic connectivity of G are also obtained. Other best possible spectral conditions for Hamiltonian s-properties are also discussed.