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We prove that the Laplacian matrix of the total graph of a finite commutative ring with identity has integer eigenvalues and present a recursive formula for computing its eigenvalues and eigenvectors. We also prove that the total graph of a finite commutative local ring with identity is super integral and give an example showing that this is not true for arbitrary rings. [ABSTRACT FROM AUTHOR]

In this paper, we initiate the study of the total zero-divisor graph over a commutative ring with unity. This graph is constructed by both relations that arise from the zero-divisor graph and from the total graph of a ring and give a joint insight of the structure of zero-divisors in a ring. We characterize Artinian rings with the connected total zero-divisor graphs and give their diameters. Moreover, we compute major characteristics of the total zero-divisor graph of the ring ℤ m of integers modulo m and prove that the total zero-divisor graphs of ℤ m and ℤ n are isomorphic if and only if m = n. [ABSTRACT FROM AUTHOR]

In this paper we study zero-divisor graphs of rings and 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 find all possible cyclic zero-divisor graphs over commutative semirings having at most one 3-cycle, and characterize all complete k-partite and regular zero-divisor graphs. Moreover, we characterize all additively cancellative commutative semirings and all commutative rings such that their zero-divisor graph has exactly one 3-cycle. [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]

Abstract: We study the idempotent matrices over a commutative antiring. We give a characterization of idempotent matrices by digraphs. We study the orbits of conjugate action and find the cardinality of orbits of basic idempotents. Finally, we prove that invertible, linear and idempotent preserving operators on matrices over entire antirings are exactly conjugate actions for . We also give a complete characterization of the case. [Copyright &y& Elsevier]