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 |H|, size e and maximum degree r such that 2e > |H|r−m. We find a best lower bound on spectral radius of graph H in terms of m and r. Let G be a connected r-regular graph of order |G| 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|G| 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|G| is odd. Our results extend the work of Cioabă, Gregory and Haemers [J. Combin. Theory Ser. B, 1999] who obtained such results for 1-factors.