共查询到10条相似文献,搜索用时 140 毫秒
1.
David F. Anderson 《Journal of Pure and Applied Algebra》2003,180(3):221-241
For a commutative ring R with set of zero-divisors Z(R), the zero-divisor graph of R is Γ(R)=Z(R)−{0}, with distinct vertices x and y adjacent if and only if xy=0. In this paper, we show that Γ(T(R)) and Γ(R) are isomorphic as graphs, where T(R) is the total quotient ring of R, and that Γ(R) is uniquely complemented if and only if either T(R) is von Neumann regular or Γ(R) is a star graph. We also investigate which cardinal numbers can arise as orders of equivalence classes (related to annihilator conditions) in a von Neumann regular ring. 相似文献
2.
The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all non-central elements of R and two distinct vertices x and y are adjacent if and only if xy = yx. Let D be a division ring and n ? 3. In this paper we investigate the diameters of Γ(Mn(D)) and determine the diameters of some induced subgraphs of Γ(Mn(D)), such as the induced subgraphs on the set of all non-scalar non-invertible, nilpotent, idempotent, and involution matrices in Mn(D). For every field F, it is shown that if Γ(Mn(F)) is a connected graph, then diam Γ(Mn(F)) ? 6. We conjecture that if Γ(Mn(F)) is a connected graph, then diam Γ(Mn(F)) ? 5. We show that if F is an algebraically closed field or n is a prime number and Γ(Mn(F)) is a connected graph, then diam Γ(Mn(F)) = 4. Finally, we present some applications to the structure of pairs of idempotents which may prove of independent interest. 相似文献
3.
Let R be a commutative ring. The total graph of R, denoted by T(Γ(R)) is a graph with all elements of R as vertices, and two distinct vertices x,y∈R, are adjacent if and only if x+y∈Z(R), where Z(R) denotes the set of zero-divisors of R. Let regular graph of R, Reg(Γ(R)), be the induced subgraph of T(Γ(R)) on the regular elements of R. Let R be a commutative Noetherian ring and Z(R) is not an ideal. In this paper we show that if T(Γ(R)) is a connected graph, then . Also, we prove that if R is a finite ring, then T(Γ(R)) is a Hamiltonian graph. Finally, we show that if S is a commutative Noetherian ring and Reg(S) is finite, then S is finite. 相似文献
4.
Wanzhou Ye 《Discrete Mathematics》2011,(21):2437
Let Fk be a mapping from RZ to RZ, satisfying that for x∈RZ and n∈Z, Fk(x)(n) is the (k+1)th largest value (median value) of the 2k+1 numbers x(n−k),…,x(n),…,x(n+k). In [3] [W.Z. Ye, L. Wang, L.G. Xu, Properties of locally convergent sequences with respect to median filter, Discrete Mathematics 309 (2009) 2775–2781], we conjectured that for k∈{2,3}, if there exists n0∈Z such that x is locally finitely convergent with respect to Fk on {n0,…,n0+k−1}, then x is finitely convergent with respect to Fk. In this paper, we obtain some sufficient conditions for a sequence finitely converging with respect to median filters. Based on these results, we prove that the conjecture is true. 相似文献
5.
Jianping Ou 《Discrete Applied Mathematics》2009,157(2):391-397
Let G be a unicyclic n-vertex graph and Z(G) be its Hosoya index, let Fn stand for the nth Fibonacci number. It is proved in this paper that Z(G)≤Fn+1+Fn−1 with the equality holding if and only if G is isomorphic to Cn, the n-vertex cycle, and that if G≠Cn then Z(G)≤Fn+1+2Fn−3 with the equality holding if and only if G=Qn or Dn, where graph Qn is obtained by pasting one endpoint of a 3-vertex path to a vertex of Cn−2 and Dn is obtained by pasting one endpoint of an (n−3)-vertex path to a vertex of C4. 相似文献
6.
Adrien Richard 《Discrete Applied Mathematics》2011,159(11):1085-1093
Given a Boolean function F:{0,1}n→{0,1}n, and a point x in {0,1}n, we represent the discrete Jacobian matrix of F at point x by a signed directed graph GF(x). We then focus on the following open problem: Is the absence of a negative circuit in GF(x) for every x in {0,1}n a sufficient condition for F to have at least one fixed point? As result, we give a positive answer to this question under the additional condition that F is non-expansive with respect to the Hamming distance. 相似文献
7.
The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R, and two distinct vertices x and y are adjacent if and only if xy = yx. The commuting graph of a group G, denoted by Γ(G), is similarly defined. In this article we investigate some graph-theoretic properties of Γ(M n (F)), where F is a field and n ≥ 2. Also we study the commuting graphs of some classical groups such as GL n (F) and SL n (F). We show that Γ(M n (F)) is a connected graph if and only if every field extension of F of degree n contains a proper intermediate field. We prove that apart from finitely many fields, a similar result is true for Γ(GL n (F)) and Γ(SL n (F)). Also we show that for two fields F and E and integers n, m ≥ 2, if Γ(M n (F))?Γ(M m (E)), then n = m and |F|=|E|. 相似文献
8.
M.R. Darafsheh 《Discrete Applied Mathematics》2009,157(4):833-837
The non-commuting graph ΓG of a non-abelian group G is defined as follows. The vertex set of ΓG is G−Z(G) where Z(G) denotes the center of G and two vertices x and y are adjacent if and only if xy≠yx. It has been conjectured that if G and H are two non-abelian finite groups such that ΓG≅ΓH, then |G|=|H| and moreover in the case that H is a simple group this implies G≅H. In this paper, our aim is to prove the first part of the conjecture for all the finite non-abelian simple groups H. Then for certain simple groups H, we show that the graph isomorphism ΓG≅ΓH implies G≅H. 相似文献
9.
Let R be a noncommutative prime ring of characteristic different from 2, let Z(R) be its center, let U be the Utumi quotient ring of R, let C be the extended centroid of R, and let f(x
1,..., x
n
) be a noncentral multilinear polynomial over C in n noncommuting variables. Denote by f(R) the set of all evaluations of f(x
1, …, xn) on R. If F and G are generalized derivations of R such that [[F(x), x], [G(y), y]] ∈ Z(R) for any x, y ∈ f(R), then one of the following holds:
(1) |
there exists α ∈ C such that F(x) = αx for all x ∈ R 相似文献
10.
Let R be a ring with center Z(R), let n be a fixed positive integer, and let I be a nonzero ideal of R. A mapping h: R → R is called n-centralizing (n-commuting) on a subset S of R if [h(x),x
n
] ∈ Z(R) ([h(x),x
n
] = 0 respectively) for all x ∈ S. The following are proved:
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