Abstract: The research on the property and structure of finite groups has always been a hot topic in group theory. It is well known that the irreducible character degree of a finite group has an important influence on the structure of the group. The authors give the relationship between the structure of groups and the degree prime-power graph, and prove that L_3\left(\textq\right) can be uniquely characterized by its order and degree prime-power graph, where q is a power of a prime and q \leqslant 31.