avila and molina [1] introduced the notion of generalized weak structures which naturally generalize minimal structures, generalized topologies and weak structures and the structures α(g),π(g),σ(g) and β (g). this work is a further investigation of generalized weak structures due to avila and molina. further we introduce the structures ro(g) and rc(g) and study the properties of the structur...

Let mGI denote the number of Laplacian eigenvalues a graph G in an interval I and let α(G) independence G. In this paper, we determine classes graphs that satisfy condition mG[0,n−α(G)]=α(G) when α(G)=2 α(G)=n−2, where n is order When α(G)=2, G≅K1∇Kn−m∇Km−1 for some m≥2. there are two types B(p,q,r) B′(p,q,r) = p + q r 2, which call binary star graphs. Also, show with determined by their spectra.

Avila and Molina [1] introduced the notion of generalized weak structures which naturally generalize minimal structures, generalized topologies and weak structures and the structures α(g),π(g),σ(g) and β(g). This work is a further investigation of generalized weak structures due to Avila and Molina. Further we introduce the structures ro(g) and rc(g) and study the properties of the structures r...

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 G be a graph with the adjacency matrix A(G), and let D(G) diagonal of degrees G. Nikiforov first defined Aα(G) as Aα(G)=αD(G)+(1−α)A(G), 0≤α≤1, which shed new light on A(G) Q(G)=D(G)+A(G), yielded some surprises. The α−adjacency energy EAα(G) is invariant that calculated from eigenvalues Aα(G). In this work, by combining theory structure properties, we provide upper lower bounds for in term...