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Anderson and Badawi (J Algebra 320(7):2706-2719, 2008) characterized all commutative rings having total graphs without any 3-cycles. In this paper we expand those results to the semiring setting and obtain the characterization of finite commutative semirings having total graphs without any 3-cycles. [ABSTRACT FROM AUTHOR]

We calculate the diameters of commuting graphs of matrices over the binary Boolean semiring, the tropical semiring and an arbitrary nonentire commutative semiring. We also find the lower bound for the diameter of the commuting graph of the semigroup of matrices over an arbitrary commutative entire antinegative semiring. [ABSTRACT FROM AUTHOR]

Let B be a nilpotent matrix and suppose that its Jordan canonical form is determined by a partition λ. Then it is known that its nilpotent commutator $ \mathcal{N}_B$ is an irreducible variety and that there is a unique partition μ such that the intersection of the orbit of nilpotent matrices corresponding to μ with $ \mathcal{N}_{B}$ is dense in $ \mathcal{N}_{B}$ . We prove that map $ \mathcal{D} $ given by $ \mathcal{D}{\left( \lambda \right)} = \mu $ is an idempotent map. This answers a question of Basili and Iarrobino [9] and gives a partial answer to a question of Panyushev [18]. In the proof, we use the fact that for a generic matrix $ A \in \mathcal{N}_{B}$ the algebra generated by A and B is a Gorenstein algebra. Thus, a generic pair of commuting nilpotent matrices generates a Gorenstein algebra. We also describe $ \mathcal{D}{\left( \lambda \right)} $ in terms of λ if $ \mathcal{D}{\left( \lambda \right)} $ has at most two parts. [ABSTRACT FROM AUTHOR]