Spectral radius of graphs with given matching number (Q869922)

In this paper, we show that of all graphs of order \(n\) with matching number \(\beta\), the graphs with maximal spectral radius are \(K_n\) if \(n = 2\beta\) or \(2\beta + 1\); \(K_{2\beta+1}\cup\overline{K_{n-2\beta-1}}\) if \(2\beta + 2 \leq n < 3\beta + 2\); \(K_\beta\vee\overline{K_{n-\beta}}\) or \(K_{2\beta+1}\cup\overline{K_{n-2\beta-1}}\) if \(n = 3\beta + 2\); \(K_\beta\vee \overline{K_{n-\beta}}\) if \(n>3\beta+2\), where \(\overline K_t\) is the empty graph on \(t\) vertices.