Name the graph class: Disjoint union of a clique and an independent set
The edge-complement of graphs in your class are complete split graphs: they can be partitioned into an independent set and a clique, such that every vertex in the independent set is adjacent to every vertex in the clique (see, for example, http://www.mathcove.net/petersen/lessons/get-lesson?les=30 ). Hence you could call your graph class co-complete split graphs.