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 byt(??(r)), is the simple (undirected) graph with vertex set r where two distinct vertices are adjacent iftheir sum lies in z(r). this work considers minimum zero-sum k-ows for t(??(r)). both for jrj evenand 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 forn = prqs (where p and q are primes, r and s are positive integers). minimum zero-sum k-ows as wellas minimum constant-sum k-ows in regular graphs are also investigated.