in this paper, we introduce a notion of property (t) for a c∗-dynamical system (a, g, α) consisting of a unital c∗-algebra a,a locally compact group g, and an action α on a. as a result,we show that if a has strong property (t) and g has kazhdan’sproperty (t), then the triple (a, g, α) has property (t).

A group is called morphic if for each normal endomorphism α in end(G),there exists β such that ker(α)= Gβ and Gα= ker(β). In this paper, we consider the case that there exist normal endomorphisms β and γ such that ker(α)= Gβ and Gα = ker(γ). We call G quasi-morphic, if this happens for any normal endomorphism α in end(G). We get the following results: G is quasi-morphic if and only if, for any ...

Let α be a coprime automorphism of group G prime order and let P an α-invariant Sylow p-subgroup G. Assume that p∉π(CG(α)). Firstly, we prove is p-nilpotent if only CNG(P)(α) centralizes P. In the case Sz(2r) PSL(2,2r)-free where r=|α|, show p-closed CG(α) normalizes As consequence these two results, obtain G≅P×H for H We also generalization Frobenius p-nilpotency theorem groups admitting autom...

Let α(G) denote the independence number of a graph G and let G ×H be the direct product of graphs G and H. Set α(G ×H) = max{α(G) · |H|, α(H) · |G|}. If G is a path or a cycle and H is a path or a cycle then α(G×H) = α(G×H). Moreover, this equality holds also in the case when G is a bipartite graph with a perfect matching and H is a traceable graph. However, for any graph G with at least one ed...

Caro, Davila, and Pepper recently proved δ(G)α(G)≤Δ(G)μ(G) for every graph G with minimum degree δ(G), maximum Δ(G), independence number α(G), matching μ(G). Answering some problems they posed, we characterize the extremal graphs δ(G)<Δ(G) as well δ(G)=Δ(G)=3.

Let B(G) = {X : X ∈ Rn×n,X = X , I ≤ X ≤ I + A(G)} and C(G) = {X : X ∈ Rn×n,X = X , I − A(G) ≤ X ≤ I + A(G)} be classes of matrices associated with graph G. Here n is the number of vertices in graph G, and A(G) is the adjacency matrix of this graph. Denote r(G) = minX∈C(G) rank(X), r+(G) = minX∈B(G) rank(X). We have shown previously that for every graph G, α(G) ≤ r+(G) ≤ χ(G) holds and α(G) = r...

in this paper we introduce the concept of α-commutator which its definition is based on generalized conjugate classes. with this notion, α-nilpotent groups, α-solvable groups, nilpotency and solvability of groups related to the automorphism are defined. n(g) and s(g) are the set of all nilpotency classes and the set of all solvability classes for the group g with respect to different automorphi...

The cardinality of a largest independent set G, denoted by α(G), is called the independence number G. domination i(G) graph G smallest dominating We introduce concept common αc(G), as greatest integer r such that every vertex belongs to some subset X VG with |X|≥r. αc(G) limit symmetry in respect fact each an and there are vertices do not belong any larger For relations between above parameters...

background: the pro-inflammatory cytokine, tumor necrosis factor-alpha (tnf-α), is a pathogenic element for a number of disorders. previous studies have reported that the -1031 t/c and -238 g/a polymorphisms in the promoter region of the tnf-α gene are important factors in reproductive-related disorders. one of the most common gynecological diseases of women during the reproductive years is end...