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.

Let M be an R-module and 0¹¦ÎM*=Hom (M, R). The graph  G¦ (M) is a graph with vertices Z¦ (M)={x ÎM \ {0} |x¦(y) = 0 or y¦(x) = 0 for some non-zero yÎM}, in which non-zero elements x and y are adjacent provided that x¦(y) = 0 or y¦ (x)=0, which introduced and studied in [3]. In this paper we associate an undirected submodule based graph G¦N (M) for each submodule N of M with vertices Z¦N (M)= {xÎM \ N | x¦(y) ÎN or y¦ (x) ÎN for some y Î M \N}, in which non-zero elements x and y are adjacent provided that x¦ (y)ÎN or y¦(x) ÎN. We observe that over a commutative ring R, G¦N (M) is connected and diam (G¦N (M)) £3. Also we get some results about clique number and connectivity number of G¦ N (M).

For an R-module M and f ∈ M∗ = Hom(M, R), let Zf(M) and Regf(M) be the sets of all zero-divisors elements and regular elements of M with re- spect to f, respectively. In this paper, we introduce the total graph of M with respect to f, denoted by T(Γf(M)), which is the graph with all the elements M as vertices, and for distinct elements m, n ∈ M, m and n are adjacent and only if m + n ∈ Zf(M). We also study the subgraphs Z(Γf(M)) and Reg(Γf(M)) with vertices Zf(M) and Regf(M), respectively.

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.

A vague graph is a generalized structure of a fuzzy graph that gives more precision, flexibility and compatibility to a system when compared with systems that are designed using fuzzy graphs. In this paper, the new concepts of (totally) irregular, strongly (totally) irregular, highly totally irregular, neighborly totally irregular, edge-irregular of vague graphs are introduced and generalized, and some properties and related results are investigated. Then, we present the concepts of homomorphism, weak isomorphism, co-weak isomorphism and isomorphism of irregular vague graphs and some results on (total) domination number, (total) 2-domination number, (total) cobondage number and (total) 2-cobondage number. Finally, one of their applications related to location map of Fire Stations and Emergency Medical centers in urban regions of a metropolis is presented.

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.

Let A be a Banach algebra and (B, B_*) be a dual Banach algebra. A linear map φ : A⟶ B is said to be a δ-homomorphism map if ‖ ‖ φ (a_1 a_2)-φ (a_1)φ (a_2)‖ ‖ ≤ δ ‖ ‖ a_1‖ ‖ ‖ ‖ a_2 ‖ ‖ for every a_1, a_2∈ A. In this paper, we study the δ-homomorphism maps from A into B. Among other things, we prove that if φ : A⟶ B is a δ-homomorphism map and B_* is multiplicative on the algebra generated by φ (A), then φ is bounded and ‖ ‖ φ ‖ ‖ ≤ 1+δ . Let A be a Banach algebra and (B, B_*) be a dual Banach algebra. A linear map φ : A⟶ B is said to be a δ-homomorphism map if ‖ ‖ φ (a_1 a_2)-φ (a_1)φ (a_2)‖ ‖ ≤ δ ‖ ‖ a_1‖ ‖ ‖ ‖ a_2 ‖ ‖ for every a_1, a_2∈ A. In this paper, we study the δ-homomorphism maps from A into B. Among other things, we prove that if φ : A⟶ B is a δ-homomorphism map and B_* is multiplicative on the algebra generated by φ (A), then φ is bounded and ‖ ‖ φ ‖ ‖ ≤ 1+δ .