Hamiltonian s-properties and eigenvalues of k-connected graphs

Chvátal and Erdös (1972) 5] proved that, for a k-connected graph G, if the stability number $\alpha (G)\le k?s$, 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 $a\ge 0$, 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 ${\lambda }_1(aD(\mathrm{\Gamma })+A(\mathrm{\Gamma }))$, 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.