On graphs whose star sets are (co-)cliques

In this paper we study graphs all of whose star sets induce cliques or co-cliques. We show that the star sets of every tree for each eigenvalue are independent sets. Among other results it is shown that each star set of a connected graph G with three distinct eigenvalues induces a clique if and only if G=K1,2 or K2,…,2. It is also proved that stars are the only graphs with three distinct eigenvalues having a star partition with independent star sets.