‎In this paper‎, ‎we introduce a class of higher homomorphisms on an algebra $ \mathcal{A} $ and we characterize the structure of them as a linear combination of some ‎sequences‎ of homomorphisms‎.‎ ‎Also ‎‎we prove that for any Approximate higher ring homomorphism on a Banach algebra $ \mathcal{A} $ under some sequences of control funtions‎, there exists a unique higher ring homomorphism near it. Using special sequences of control functions, we show that the Approximate higher ring homomorphism is an exact higher ring homomorphism.

In this paper, we prove that every orthogonally higher ring derivation is a higher ring derivation. Also we find the general solution of the pexider orthogonally higher ring derivations {fn(x + y) = gn(x) + hn(y), = 0, {fn(xy) =∑, i+j=n gi (x)hj (y). Then we prove that for any Approximate pexider orthogonally higher ring derivation under some control functions φ, (x, y) and ψ, (x, y), there exists a unique higher ring derivation D = {dn}∞,n=0, near {fn}∞,n=0, {gn}∞,n=0 and {hn}∞,n=0 estimated by φ,and ψ, .

In this paper, we de ne some new homomorphism-monotone parameters for hy-pergraphs. Using these parameters, we extend some graph homomorphism results to hyper-graph case. Also, we present some bounds for some well-known invariants of hypergraphs such as fractional chromatic number, independent numer and some other invariants of hyergraphs, in terms of these parameters.

In this paper we introduce the notion of φ-commutativity for a Banach algebra A, where φ is a continuous homomorphism on A and study the concept of φ-weak amenability for φ-commutative Banach algebras. We give an example to show that the class of φ-weakly amenable Banach algebras is larger than that of weakly amenable commutative Banach algebras. We characterize φ-weak amenability of φ-commutative Banach algebras and prove some hereditary properties. Moreover we verify some of the previous available results about commutative weakly amenable Banach algebras, for φ-commutative φ-weakly amenable Banach algebras.

The clique polynomial of a graph G is the ordinary generating function of the number of complete subgraphs (cliques) of G. In this paper, we introduce a new vertex-weighted version of these polynomials. We also show that these weighted clique polynomials have always a real root provided that the weights are non-negative real numbers. As an application, we obtain a no-homomorphism criteria based on the largest real root of our vertex-weighted clique polynomial.