Minimum flows in the total graph of a finite commutative ring

Abstract

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 are‎ ‎adjacent 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 $k$-flows‎ ‎as well as minimum constant-sum $k$-flows in regular graphs are also investigated‎.