A new characterization of some characteristically simple groups

Let G be a finite group and cd(G) be the set of irreducible complex character degrees of G. It was proved that some finite simple groups are uniquely determined by their orders and their degree graphs. Recently, in [Behravesh, et al., Recognition of Janko groups and some simple K4-groups by the order and one irreducible character degree or character degree graph, Int. J. Group Theory, DOI: 10.22108/ijgt.2019.113029.1502.] new characterizations for some finite simple groups are given. Also, in [Qin, et al., Mathieu groups, and its degree prime-power graphs, Comm. Algebra, 2019] the degree prime-power graph of a finite group is introduced and it is proved that the Mathieu groups are uniquely determined by order and degree prime-power graph. In this paper, we continue this work and we characterize some simple groups and some characteristically simple groups by their orders and some vertices of their degree prime-power graphs.