In computer science and mathematics, a partition of a set into two or more disjoint subsets with equal sums is a well-known NP-complete problem. This is a hard problem and referred to as the partition problem or number partitioning. In this paper, we solve a particular type of NP-complete problem on the set of all zero-divisors of Z n including zero, where Z n is the ring of residue classes of a positive integer n . In this regard, we introduce and investigate quadratic zero-divisor graph in which we build an edge between zero-divisors z i and z j if and only if z i 2 ≡ z j 2 mod n , i ≠ j . This is denoted as G ⏞ 2 , n . We characterize these graphs in term of complete graphs for classes of integers 2 α , p α , 2 α p , 2 p α and p q , where α is any positive integer and p , q are odd primes.