The Zero-divisor Graphs of Semirings
نویسندگان
چکیده
In this paper we study zero-divisor graphs of semirings. We show that all zero-divisor graphs of (possibly noncommutative) semirings are connected and have diameter less than or equal to 3. We characterize all acyclic zero-divisor graphs of semirings and prove that in the case zero-divisor graphs are cyclic, their girths are less than or equal to 4. We also give a description of the zero-divisor graphs over commutative semirings with girth equal to 4 and the ones with girth equal to 3 and only one 3-cycle. Moreover, we characterize all additively cancellative semirings and all rings such that their zero-divisor graph has exactly one 3-cycle.
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