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The non-commuting graph $Gamma_R$ of a finite ring $R$ with center $Z(R)$ is a simple undirected graph whose vertex set is $R setminus Z(R)$ and two distinct vertices $a$ and $b$ are adjacent if and only if $ab e ba$. In this paper, we show that $Gamma_R$ is not isomorphic to certain graphs of any finite non-commutative ring $R$. Some connections between $Gamma_R$ and commuting probability of $R$ are also obtained. Further, it is shown that the non-commuting graphs of two $mathbb{Z}$-isoclinic rings are isomorphic if the centers of the rings have same order
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In this paper we deal with the problem of computing the sum of the $k$-th powers of all the elements of the matrix ring $mathbb{M}_d(R)$ with $d>1$ and $R$ a finite commutative ring. We completely solve the problem in the case $R=mathbb{Z}/nmathbb{Z}$ and give some results that compute the value of this sum if $R$ is an arbitrary finite commutative ring $R$ for many values of $k$ and $d$. Finally, based on computational evidence and using some technical results proved in the paper we conjecture that the sum of the $k$-th powers of all the elements of the matrix ring $mathbb{M}_d(R)$ is always $0$ unless $d=2$, $textrm{card}(R) equiv 2 pmod 4$, $1<kequiv -1,0,1 pmod 6$ and the only element $ein R setminus {0}$ such that $2e =0$ is idempotent, in which case the sum is $textrm{diag}(e,e)$.
The prime graph question asks whether the Gruenberg-Kegel graph of an integral group ring $mathbb Z G$ , i.e. the prime graph of the normalised unit group of $mathbb Z G$ coincides with that one of the group $G$. In this note we prove for finite groups $G$ a reduction of the prime graph question to almost simple groups. We apply this reduction to finite groups $G$ whose order is divisible by at most three primes and show that the Gruenberg - Kegel graph of such groups coincides with the prime graph of $G$.
This work relates to three problems, the classification of maximal Abelian subalgebras (MASAs) of the Lie algebra of square matrices, the classification of 2-step solvable Frobenius Lie algebras and the Gerstenhabers Theorem. Let M and N be two commuting square matrices of order n with entries in an algebraically closed field K. Then the associative commutative K-algebra, they generate, is of dimension at most n. This result was proved by Murray Gerstenhaber in 1961. The analog of this property for three commuting matrices is still an open problem, its version for a higher number of commuting matrices is not true in general. In the present paper, we give a sufficient condition for this property to be satisfied, for any number of commuting matrices and arbitrary field K. Such a result is derived from a discussion on the structure of 2-step solvable Frobenius Lie algebras and a complete characterization of their associated left symmetric algebra structure. We discuss the classification of 2-step solvable Frobenius Lie algebras and show that it is equivalent to that of n-dimensional MASAs of the Lie algebra of square matrices, admitting an open orbit for the contragradient action associated to the multiplication of matrices and vectors. Numerous examples are discussed in any dimension and a complete classification list is supplied in low dimensions. Furthermore, in any finite dimension, we give a full classification of all 2-step solvable Frobenius Lie algebras corresponding to nonderogatory matrices.
The commuting graph of a group $G$ is the simple undirected graph whose vertices are the non-central elements of $G$ and two distinct vertices are adjacent if and only if they commute. It is conjectured by Jafarzadeh and Iranmanesh that there is a universal upper bound on the diameter of the commuting graphs of finite groups when the commuting graph is connected. In this paper we determine upper bounds on the diameter of the commuting graph for some classes of groups to rule them out as possible counterexamples to this conjecture. We also give an example of an infinite family of groups with trivial centre and diameter 6, the previously largest known diameter for an infinite family was 5 for $S_n$.
This paper studies commuting matrices in max algebra and nonnegative linear algebra. Our starting point is the existence of a common eigenvector, which directly leads to max analogues of some classical results for complex matrices. We also investigate Frobenius normal forms of commuting matrices, particularly when the Perron roots of the components are distinct. For the case of max algebra, we show how the intersection of eigencones of commuting matrices can be described, and we consider connections with Boolean algebra which enables us to prove that two commuting irreducible matrices in max algebra have a common eigennode.
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