Minimum flows in the total graph of a finite commutative ring

Document Type : Research Paper

Authors

1 Ostfalia Hochschule fur Angewandte Wissenschaften

2 Palestine Technical University

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.

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References

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
S. Akbari, N. Ghareghani, G. B. Khosrovshahi and A. Mahmoody (2009). On zero-sum 6- ows of graphs. Linear Algebra Appl.. 430, 3047-3052
S. Akbari, A. Daemi, O. Hatami, A. Javanmard and A. Mehrabian (2010). Zero-sum ows in regular graphs. Graphs Combin.. 26, 603-615
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
D. F. Anderson and A. Badawi (2008). The total graph of a commutative ring. J. Algebra,. 320, 2706-2719
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
T. T. Chelvam and T. Asir (2011). Domination in the total graph on Zn. Discrete Math. Algorithms Appl.. 3, 413-421
T. T. Chelvam and T. Asir (2013). Domination in the total graph of a commutative ring. J. Combin. Math. Combin. Comput.. 87, 147-158
T. T. Chelvam and T. Asir (2011). A note on total graph of Zn. J. Discrete Math. Sci. Cryptogr.. 14, 1-7
D. Konig (1916). Uber Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre. (German), Math. Ann.. 77, 453-465
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
J. Petersen (1891). Die Theorie der regularen Graphs. Acta Math.. 15, 193-220
H. Shahmohamad (2002). On minimum ow number of graphs. Bull. Inst. Combin. Appl.. 35, 26-36
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
C. Q. Zang (1997). Integer ows and cycle covers of graphs. Marcel Dekker, Marcel Dekker,.
Transactions on Combinatorics

Volume 3, Issue 3 - Serial Number 3
September 2014
Pages 11-20

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