Regular factors of regular graphs from eigenvalues

Abstract

Let r and m be two integers such that r ≥ m. Let H be a graph with order {pipe}H{pipe}, size e and maximum degree r such that 2e ≥ {pipe}H{pipe}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 {pipe}G{pipe} 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{pipe}G{pipe} 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{pipe}G{pipe} 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.