Polona Oblak

THE ZERO-DIVISOR GRAPHS OF RINGS AND SEMIRINGS

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.

On maximal distances in a commuting graph

We study maximal distances in the commuting graphs of matrix algebras defined over algebraically closed fields. In particular, we show that the maximal distance can be attained only between two nonderogatory matrices. We also describe rank-one and semisimple matrices using the distances in the commuting graph.

The upper bound for the index of nilpotency for a matrix commuting with a given nilpotent matrix

We study the set ( ) of all possible Jordan canonical forms of nilpotent matrices commuting with a given nilpotent matrix B. We describe ( ) in the special case when B has only one Jordan block and discuss some consequences. In the general case, we find the maximal possible index of nilpotency in the set of all nilpotent matrices commuting with a given nilpotent matrix. We consider several examples.

Commuting graphs and extremal centralizers

We determine the conditions for matrix centralizers which can guarantee the connectedness of the commuting graph for the full matrix algebra M n ( F ) over an arbitrary field F . It is known that if F is an algebraically closed field and n  ≥ 3 , then the diameter of the commuting graph of M n ( F ) is always equal to four. We construct a concrete example showing that if F is not algebraically closed, then the commuting graph of M n ( F ) can be connected with the diameter at least five.

On the nilpotent commutator of a nilpotent matrix

We study the structure of the nilpotent commutator 𝒩 B of a nilpotent matrix B. We show that 𝒩 B intersects all nilpotent orbits for conjugation if and only if B is a square-zero matrix. We describe nonempty intersections of 𝒩 B with nilpotent orbits in the case the n × n matrix B has rank n − 2. Moreover, we give some results concerning the inverse image of the map taking B to the maximal nilpotent orbit intersecting 𝒩 B .

On diameter of the commuting graph of a full matrix algebra over a finite field

It is shown that the commuting graph of a matrix algebra over a finite field has diameter at most five if the size of the matrices is not a prime nor a square of a prime. It is further shown that the commuting graph of even-sized matrices over finite field has diameter exactly four. This partially proves a conjecture stated by Akbari, Mohammadian, Radjavi, and Raja Linear Algebra Appl. 418 (2006) 161-176.

The Total Graphs of Finite Rings

In this paper we extend the study of total graphs τ(R) to noncommutative finite rings R. We prove that τ(R) is connected if and only if R is not local, and we see that in that case τ(R) is always Hamiltonian. We also find an upper bound for the domination number of τ(R) for all finite rings R.

Cholesky decomposition of matrices over commutative semirings

ABSTRACT We prove that over a commutative semiring every symmetric strongly invertible matrix with nonnegative numerical range has a Cholesky decomposition.

Diameters of conneted components of commuting graphs

In this paper, diameters of connected components of commuting graphs of GLn(S) are calculated, for an integer n ≥ 2 and a commutative antinegative entire semiring S, unless n is a non-prime odd number and S has at least two invertible elements.