@article {
author = {Sander, Torsten and Nazzal, Khalida},
title = {Minimum flows in the total graph of a finite commutative ring},
journal = {Transactions on Combinatorics},
volume = {3},
number = {3},
pages = {11-20},
year = {2014},
publisher = {University of Isfahan},
issn = {2251-8657},
eissn = {2251-8665},
doi = {10.22108/toc.2014.5252},
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 $\vert R\vert$ 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.},
keywords = {Constant-sum k-flow,minimum flow,the ring of integers modulo n,total graph of a commutative ring,zero-sum k-flow},
url = {https://toc.ui.ac.ir/article_5252.html},
eprint = {https://toc.ui.ac.ir/article_5252_34bd2a18fa39bc58c69e5a7037b0e84c.pdf}
}