This　paper　considers　the　leader-following　consensus　problem　for　nonlinear　multiagent　systems　with　large　delays.　The　time　delays　switch　alternately　between　two　different　kinds　of　delays,　called　a　small　delay　and　a　large　delay.　New　Lyapunov　functionals　that　depend　on　graph　information　are　constructed.　To　guarantee　the　desired　leader-following　consensus,　the　concepts　of　the　length　and　frequency　of　the　large　delay　periods　are　introduced.　Sufficient　conditions　for　consensus　are　given　in　terms　of　linear　matrix　inequalities.　Furthermore,　low-dimensional　criteria　that　are　easily　implemented　especially　for　large-scale　multiagent　systems　are　provided.　Finally,　two　simulation　examples　are　presented　to　validate　the　effectiveness　of　the　proposed　scheme.