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
Article 2, Volume 3, Issue 3, September 2014, Page 11-20	 PDF (234.45 K)
Document Type: Research Paper
Torsten Sander1; Khalida Mohammad Nazzal 2
1Ostfalia Hochschule fur Angewandte Wissenschaften
2Palestine Technical University
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.
Constant-sum k-flow; minimum flow; the ring of integers modulo n; total graph of a commutative ring; zero-sum k-flow
05C21 Flows in graphs; 05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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