Minimum flows in the total graph of a finite commutative ring

Let R be a commutative ring with zero-divisor set Z(R). The total graph of R, denoted by T(􀀀(R)),is the simple (undirected) graph with vertex set R, and two distinct vertices are adjacent if their sum is a zero divisor in R. Let T(􀀀(Zn)) be the total graph for the ring of integers modulo n. Minimum zero-sum k-flows for T(􀀀(Zn)) are constructed, in particular, the cases n is even, or n = p^r, n = p^rq^s, where p and q are primes, r and s are positive integers, are considered. Minimum zero-sum k-flows as well as minimum constant-sum k-flows in regular graphs are also investigated.