Paper Information

Journal:   TRANSACTIONS ON COMBINATORICS   2014 , Volume 3 , Number 3; Page(s) 11 To 20.
 
Paper: 

MINIMUM FLOWS IN THE TOTAL GRAPH OF A FINITE COMMUTATIVE RING

 
 
Author(s):  SANDER T., NAZZAL K.*
 
* DEPARTMENT OF MATHEMATICS, PALESTINE TECHNICAL UNIVERSITY -KADOORIE, TULKARM, WEST BANK, PALESTINE
 
Abstract: 

Let R be a commutative ring with zero-divisor set Z (R). The total graph of R, denoted by T(G(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(G(R)). Both for |R| even and the case when |R| is odd and Z(G) is an ideal of R it is shown that T(G(R)) has a zero-sum 3-flow, but no zero-sum 2-flow. As a step towards resolving the remaining case, the total graphT(G(Z n)) for the ring of integers modulo n is considered. Here, minimum zero-sum k-flows are obtained for n=prqs (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.

 
Keyword(s): TOTAL GRAPH, CONSTANT-SUM K-FLOW, ZERO-SUM FLOW, MINIMUM FLOW
 
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