Minimum Flows in the Total Graph of a Commutative Ring

Abstract

Let R be a commutative ring with zero-divisor set Z(R)‎. ‎The total graph of R‎, ‎denoted by‎ ‎T(Γ(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(Γ(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(Γ(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(Γ(Zn))‎ ‎for the ring of integers modulo n is considered‎. ‎Here‎, ‎minimum zero-sum k-flows are obtained for n=pr and 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.