Regular factors of regular graphs from eigenvalues

Let r and m be two integers such that r >= m. Let H be a graph with order vertical bar H vertical bar, size e and maximum degree r such that 2e >= vertical bar H vertical bar r-m.We find a best lower bound on spectra lradius of graph H in terms of m and r. Let G be a connected r-regular graph of order vertical bar G vertical bar and k < r be an integer. Using the previous results, we find some best upper bounds (in terms of r and k) on the third largest eigenvalue that is sufficient to guarantee that G has a k-factor when k vertical bar G vertical bar is even. Moreover, we find a best bound on the second largest eigenvalue that is sufficient to guarantee that G is k-critical when k vertical bar G vertical bar is odd. Our results extend the work of Cioaba, Gregory and Haemers [J.Combin. Theory Ser. B, 1999] who obtained such results for 1-factors.