Let G be a , nite group and cd(G) be the set of irreducible complex character degrees of G. It was proved that some , nite 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 , nite 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 , nite 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.

The set of element orders of a nite group G is called the spectrum. GROUPS with coinciding spectra are said to be isospectral. It is known that if G has a nontrivial normal soluble subgroup then there exist in nitely many pairwise non-isomorphic GROUPS isospectral to G. The situation is quite different if G is a nonabelain SIMPLE group. Recently it was proved that ifL is a SIMPLE classical group of dimension at least 62 and G is a nite group isospectral to L, then up to isomorphism L£G£ Aut L. We show that the assertion remains true if 62 is replaced by 38.

For a group G, π, e(G) and sm(G) are denoted the set of orders of elements and the number of elements of order m in G, respectively. Let nse(G) = fsm(G) j m 2 π, e(G)g. An arbitrary , nite group M is NSE characterization if, for every group G, the equality nse(G) = nse(M) implies that G , = M. In this paper, we are going to show that the non-Abelian finite SIMPLE GROUPS A9, A10, A12, U4(3), U5(2), U6(2), S6(2), O+ 8 (2) and HS are characterizable by NSE.