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 where two distinct vertices are adjacent if their sum lies in Z(R). This work considers minimum zero-sum k- ows for T(??(R)). Both for jRj even and the case when jRj is odd and Z(G) is an ideal of R it is shown that T(??(R)) has a zero-sum 3- ow, but no zero-sum 2- ow. As a step towards resolving the remaining case, the total graph T(??(Zn)) for the ring of integers modulo n is considered. Here, minimum zero-sum k- ows are obtained for n = prqs (where p and q are primes, r and s are positive integers). Minimum zero-sum k- ows as well as minimum constant-sum k- ows in regular graphs are also investigated.