Journal of Mathematical Extension
Volume:12 Issue: 1, Winter 2018

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.

All finite groups with toroidal or projective cyclic graphs are classified. Indeed, it is shown that the only finite groups with projective cyclic graphs are S3 × Z2, D14, QD16 and <x, y : x^7 = y^3 = 1; x^y = x^2> which all have toroidal cyclic graph too. Also, D16 is characterized as the only finite group whose cyclic graph is toroidal but not projective

In this paper, we consider a nonsmooth multiobjective programming problem with equilibrium constraints. We present three constraint qualications (CQs) and investigate their relations. Furthermore, we derive two types of necessary optimality conditions under these CQs. In addition, some examples are given to clarify our result

Grouplikes have been introduced and studied by the first author.A grouplike is something between semigroup and group and its axioms are generalizationof the four group axioms. We observe that every grouplike is a homogroup (a semigroupcontaining an ideal subgroup) with a unique central idempotent.On the other hand, decomposer and associative functions on groups, semigroupsand even magmas are introduced in 2007.If $(G,\cdot)$ is a group and $f:G\rightarrow G$ is an associative function (i.e.$f(xf(yz))=f(f(xy)z)$, for all $x,y,z\in G$), then the $f$-multiplication "$\cdot_f$"defined by $x\cdot_f y =f(xy)$, is an associative binary operation with severalinteresting properties. A nice example for associative function,$f$-multiplication and such algebraic structures are $b$-decimal part functions$(\; )_b$, $b$-addition $+_b$, and the real $b$-grouplike $(\mathbb{R},+_b)$.In this paper, we introduce an important type of grouplikes (namely $f$-grouplike) that is motivated fromthe both topics. We prove that $f$-grouplikes is a proper subclass of Class United Grouplikes, study some of their properties and show some of future directionsfor the researches.

In this ‎paper,‎ we show that the relative algebraic interior is a suitable replacement for both of the topological interior and the algebraic interior for the cases where these are empty‎. ‎Also‎, ‎we presente ‎some ‎properties ‎of ‎(‎relative)‎ algebraic ‎interior ‎and‎ ‎some fixed point theorems for increasing mapping‎. ‎The results obtained can be viewed as an extension and improvement of ‎the known corresponding results‎‎. ‎Some examples are provided here to support our conclusions‎.

The Smaradanche's neutrosophic set theory has become a popular topic of investigation in the fuzzyand intuitionistic community. However, there is less investigation on the cut sets and extension principlesfor neutrosophic sets and neutrosophic multi-sets, as well as algebraic operations. In this paper, we first proposed the extension principles of neutrosophic multi-sets and cut sets which is a bridge betweenneutrosophic multi-sets and crisp sets. Then the representation theorem of neutrosophic multi-setsbased on cut sets are studied. Finally, the addition, subtraction, multiplication and division operationsover neutrosophic multi-sets are defined based on the extension principle.

The concept of Γ-semihyperrings is a generalization of a semiring, a generalization of a Γ-semiring, and a generalization of a semihyperering. In this paper, we define the notions of complex product, extension property and flat Γ-semihyperrings and some of their properties are obtained. In addition, we prove that every flat Γ-semihyperring is absolutely extendable. Finally, we give some characterization of stable elements.

he hybrid fuzzy differential equations (HFDEs) are natural way to model dynamic systems with embedded uncertainty and have a wide range of applications in science and engineering that make them useful. The present study is an attempt to obtain the numerical solutions of hybrid fuzzy differential equations through homotopy perturbation method (HPM). The convergence of the HPM to solve hybrid fuzzy differential equations is investigated in detail in this paper. In addition, the validity and efficiency of the proposed method is investigated and verified by several numerical examples.