Spectral radius of graphs with given matching number

In this paper, we show that of all graphs of order n with matching number β, the graphs with maximal spectral radius are Kn if n = 2β or 2β + 1; if 2β + 2 ⩽ n < 3β + 2; or if n = 3β + 2; if n > 3β + 2, where is the empty graph on t vertices.