Sander, T., Nazzal, K. (2014). Minimum flows in the total graph of a finite commutative ring. Transactions on Combinatorics, 3(3), 11-20. doi: 10.22108/toc.2014.5252

Torsten Sander; Khalida Mohammad Nazzal. "Minimum flows in the total graph of a finite commutative ring". Transactions on Combinatorics, 3, 3, 2014, 11-20. doi: 10.22108/toc.2014.5252

Sander, T., Nazzal, K. (2014). 'Minimum flows in the total graph of a finite commutative ring', Transactions on Combinatorics, 3(3), pp. 11-20. doi: 10.22108/toc.2014.5252

Sander, T., Nazzal, K. Minimum flows in the total graph of a finite commutative ring. Transactions on Combinatorics, 2014; 3(3): 11-20. doi: 10.22108/toc.2014.5252

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 are adjacent if their sum lies in $Z(R)$. This work considers minimum zero-sum $k$-flows for $T(\Gamma(R))$. Both for $\vert R\vert$ even and the case when $\vert R\vert$ 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.

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