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$t(gamma(r))$, is the simple (undirected) graph with vertex set  $r$ where two distinct vertices areadjacent if their sum lies in $z(r)$.this work considers minimum zero-sum $k$-flows for $t(gamma(r))$.both for $vert rvert$ even and the case when $vert rvert$ is odd and $z(g)$ is an ideal of $r$it is shown that $t(gamma(r))$ has a zero-sum $3$-flow, but no zero-sum $2$-flow.as a step towards resolving the remaining case, the total graph $t(gamma(mathbb{z}_n ))$for the ring of integers modulo $n$ is considered. here,minimum zero-sum $k$-flows are obtained for $n = p^r$ and $n = p^r q^s$ (where $p$and $q$ are primes, $r$ and $s$ are positive integers).minimum zero-sum $k$-flowsas well as minimum constant-sum $k$-flows in regular graphs are also investigated.