The degree pattern of a finite group G associated with its prime graph has been introduced by Moghaddamfar in 2005 and it is proved that the following simple groups are uniquely determined by their order and degree patterns： All sporadic simple groups, the alternating groups Ap （p ≤ 5 is a twin prime） and some simple groups of the Lie type. In this paper, the authors continue this investigation. In particular, the authors show that the symmetric groups Sp＋3, where p ＋ 2 is a composite number and p ＋ 4 is a prime and 97 〈 p ∈π（1000!）, are 3-fold OD-characterizable. The authors also show that the alternating groups All6 and A134 are OD-characterizable. It is worth mentioning that the latter not only generalizes the results by Hoseini in 2010 but also gives a positive answer to a conjecture by Moghaddamfar in 2009.
