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1.
Let Mn be the space of all n × n complex matrices, and let Γn be the subset of Mn consisting of all n × n k-potent matrices. We denote by Ψn the set of all maps on Mn satisfying A − λB ∈ Γn if and only if ?(A) − λ?(B) ∈ Γn for every A,B ∈ Mn and λ ∈ C. It was shown that ? ∈ Ψn if and only if there exist an invertible matrix P ∈ Mn and c ∈ C with ck−1 = 1 such that either ?(A) = cPAP−1 for every A ∈ Mn, or ?(A) = cPATP−1 for every A ∈ Mn.  相似文献   

2.
By the signless Laplacian of a (simple) graph G we mean the matrix Q(G)=D(G)+A(G), where A(G),D(G) denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of G. It is known that connected graphs G that maximize the signless Laplacian spectral radius ρ(Q(G)) over all connected graphs with given numbers of vertices and edges are (degree) maximal. For a maximal graph G with n vertices and r distinct vertex degrees δr>δr-1>?>δ1, it is proved that ρ(Q(G))<ρ(Q(H)) for some maximal graph H with n+1 (respectively, n) vertices and the same number of edges as G if either G has precisely two dominating vertices or there exists an integer such that δi+δr+1-i?n+1 (respectively, δi+δr+1-i?δl+δr-l+1). Graphs that maximize ρ(Q(G)) over the class of graphs with m edges and m-k vertices, for k=0,1,2,3, are completely determined.  相似文献   

3.
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   

4.
Let F be a field with ∣F∣ > 2 and Tn(F) be the set of all n × n upper triangular matrices, where n ? 2. Let k ? 2 be a given integer. A k-tuple of matrices A1, …, Ak ∈ Tn(F) is called rank reverse permutable if rank(A1 A2 ? Ak) = rank(Ak Ak−1 ? A1). We characterize the linear maps on Tn(F) that strongly preserve the set of rank reverse permutable matrix k-tuples.  相似文献   

5.
We prove that if Γ is a finite connected graph having the same convex subgraphs as the graph of the n-dimensional cube Qn (n?3), then |V(Γ)|?|V(Qn)|. Moreover, if |V(Γ)|=|V(Qn)|, Γ is isomorphic to Qn.  相似文献   

6.
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   

7.
8.
By the signless Laplacian of a (simple) graph G we mean the matrix Q(G)=D(G)+A(G), where A(G),D(G) denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of G. For every pair of positive integers n,k, it is proved that if 3?k?n-3, then Hn,k, the graph obtained from the star K1,n-1 by joining a vertex of degree 1 to k+1 other vertices of degree 1, is the unique connected graph that maximizes the largest signless Laplacian eigenvalue over all connected graphs with n vertices and n+k edges.  相似文献   

9.
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 are joint by an edge whenever they commute. It is conjectured that if R is a ring with identity such that Γ(R) ≈ Γ(M n (F)), for a finite field F and n ≥ 2, then RM n (F). Here we prove this conjecture when n = 2.  相似文献   

10.
Let F be a field and let m and n be integers with m,n?3. Let Mn denote the algebra of n×n matrices over F. In this note, we characterize mappings ψ:MnMm that satisfy one of the following conditions:
1.
|F|=2 or |F|>n+1, and ψ(adj(A+αB))=adj(ψ(A)+αψ(B)) for all A,BMn and αF with ψ(In)≠0.
2.
ψ is surjective and ψ(adj(A-B))=adj(ψ(A)-ψ(B)) for every A,BMn.
Here, adjA denotes the classical adjoint of the matrix A, and In is the identity matrix of order n. We give examples showing the indispensability of the assumption ψ(In)≠0 in our results.  相似文献   

11.
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,yR, are adjacent if and only if x+yZ(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   

12.
Let G=(V,E) be a graph with V={1,2,…,n}. Define S(G) as the set of all n×n real-valued symmetric matrices A=[aij] with aij≠0,ij if and only if ijE. By M(G) we denote the largest possible nullity of any matrix AS(G). The path cover number of a graph G, denoted P(G), is the minimum number of vertex disjoint paths occurring as induced subgraphs of G which cover all the vertices of G.There has been some success with relating the path cover number of a graph to its maximum nullity. Johnson and Duarte [5], have shown that for a tree T,M(T)=P(T). Barioli et al. [2], show that for a unicyclic graph G,M(G)=P(G) or M(G)=P(G)-1. Notice that both families of graphs are outerplanar. We show that for any outerplanar graph G,M(G)?P(G). Further we show that for any partial 2-path G,M(G)=P(G).  相似文献   

13.
Let G be a graph with n vertices and m edges and let μ(G) = μ1(G) ? ? ? μn(G) be the eigenvalues of its adjacency matrix. Set s(G)=∑uV(G)d(u)-2m/n∣. We prove that
  相似文献   

14.
Let F(A) be the numerical range or the numerical radius of a square matrix A. Denote by A ° B the Schur product of two matrices A and B. Characterizations are given for mappings on square matrices satisfying F(A ° B) = F(?(A) ° ?(B)) for all matrices A and B. Analogous results are obtained for mappings on Hermitian matrices.  相似文献   

15.
Let G be a graph whose Laplacian eigenvalues are 0 = λ1 ? λ2 ? ? ? λn. We investigate the gap (expressed either as a difference or as a ratio) between the extremal non-trivial Laplacian eigenvalues of a connected graph (that is λn and λ2). This gap is closely related to the average density of cuts in a graph. We focus here on the problem of bounding the gap from below.  相似文献   

16.
Denote by An the set of square (0, 1) matrices of order n. The set An, n ? 8, is partitioned into row/column permutation equivalence classes enabling derivation of various facts by simple counting. For example, the number of regular (0, 1) matrices of order 8 is 10160459763342013440. Let Dn, Sn denote the set of absolute determinant values and Smith normal forms of matrices from An. Denote by an the smallest integer not in Dn. The sets D9 and S9 are obtained; especially, a9 = 103. The lower bounds for an, 10 ? n ? 19 (exceeding the known lower bound an ? 2fn − 1, where fn is nth Fibonacci number) are obtained. Row/permutation equivalence classes of An correspond to bipartite graphs with n black and n white vertices, and so the other applications of the classification are possible.  相似文献   

17.
Let Mn(R) be the algebra of all n×n matrices over a unital commutative ring R with 2 invertible, V be an R-module. It is shown in this article that, if a symmetric bilinear map {·,·} from Mn(RMn(R) to V satisfies the condition that {u,u}={e,u} whenever u2=u, then there exists a linear map f from Mn(R) to V such that . Applying the main result we prove that an invertible linear transformation θ on Mn(R) preserves idempotent matrices if and only if it is a Jordan automorphism, and a linear transformation δ on Mn(R) is a Jordan derivation if and only if it is Jordan derivable at all idempotent points.  相似文献   

18.
We show that every injective Jordan semi-triple map on the algebra Mn(F) of all n × n matrices with entries in a field F (i.e. a map Φ:Mn(F)→Mn(F) satisfying
Φ(ABA)=Φ(A)Φ(B)Φ(A)  相似文献   

19.
Let Mn be the algebra of all n×n matrix over a field F, A a rank one matrix in Mn. In this article it is shown that if a bilinear map ? from Mn×Mn to Mn satisfies the condition that ?(u,v)=?(I,A) whenever u·v=A, then there exists a linear map φ from Mn to Mn such that . If ? is further assumed to be symmetric then there exists a matrix B such that ?(x,y)=tr(xy)B for all x,yMn. Applying the main result we prove that if a linear map on Mn is desirable at a rank one matrix then it is a derivation, and if an invertible linear map on Mn is automorphisable at a rank one matrix then it is an automorphism. In other words, each rank one matrix in Mn is an all-desirable point and an all-automorphisable point, respectively.  相似文献   

20.
We study determinant inequalities for certain Toeplitz-like matrices over C. For fixed n and N ? 1, let Q be the n × (n + N − 1) zero-one Toeplitz matrix with Qij = 1 for 0 ? j − i ? N − 1 and Qij = 0 otherwise. We prove that det(QQ) is the minimum of det(RR) over all complex matrices R with the same dimensions as Q satisfying ∣Rij∣ ? 1 whenever Qij = 1 and Rij = 0 otherwise. Although R has a Toeplitz-like band structure, it is not required to be actually Toeplitz. Our proof involves Alexandrov’s inequality for polarized determinants and its generalizations. This problem is motivated by Littlewood’s conjecture on the minimum 1-norm of N-term exponential sums on the unit circle. We also discuss polarized Bazin-Reiss-Picquet identities, some connections with k-tree enumeration, and analogous conjectured inequalities for the elementary symmetric functions of QQ.  相似文献   

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