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

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

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

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

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.

S. Akbari, D. Kiani, F. Mohammadi and S. Moradi (2009). The total graph and regular graph of a commutative ring. J.
Pure Appl. Algebra. 213, 2224-2228

2

S. Akbari, N. Ghareghani, G. B. Khosrovshahi and A. Mahmoody (2009). On zero-sum 6-
ows of graphs. Linear Algebra
Appl.. 430, 3047-3052

3

S. Akbari, A. Daemi, O. Hatami, A. Javanmard and A. Mehrabian (2010). Zero-sum
ows in regular graphs. Graphs
Combin.. 26, 603-615

4

S. Akbari, N. Ghareghani, G. B. Khosrovshahi and S. Zare (2012). A note on zero-sum 5-
ows in regular graphs. Electron.
J. Combin., Paper 7. 19, 0

5

D. F. Anderson and A. Badawi (2008). The total graph of a commutative ring. J. Algebra,. 320, 2706-2719

6

D. F. Anderson and A. Badawi (2012). The total graph of a commutative ring without the zero element. J. Algebra Appl., DOI: 10.1142/S0219498812500740. 11, 0

7

T. T. Chelvam and T. Asir (2011). Domination in the total graph on Zn. Discrete Math. Algorithms Appl.. 3, 413-421

8

T. T. Chelvam and T. Asir (2013). Domination in the total graph of a commutative ring. J. Combin. Math. Combin.
Comput.. 87, 147-158

9

T. T. Chelvam and T. Asir (2011). A note on total graph of Zn. J. Discrete Math. Sci. Cryptogr.. 14, 1-7

10

D. Konig (1916). Uber Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre. (German), Math. Ann.. 77, 453-465

11

H. R. Maimani, C. Wickham and S. Yassemi (2012). Rings whose total graphs have genus at most one. Rocky Mountain
J. Math.. 42, 1551-1560

12

J. Petersen (1891). Die Theorie der regularen Graphs. Acta Math.. 15, 193-220

13

H. Shahmohamad (2002). On minimum
ow number of graphs. Bull. Inst. Combin. Appl.. 35, 26-36

14

M. H. Shekarriz, M. H. Shirdareh Haghighi and H. Sharif (2012). On the total graph of a nite commutative ring. Comm.
Algebra. 40, 2798-2807

15

C. Q. Zang (1997). Integer
ows and cycle covers of graphs. Marcel Dekker, Marcel Dekker,.