Linguistic variables play a vital role in several qualitative decision environments, in which decision-makers assume several feasible linguistic values or criteria instead of a single term for an alternative or variable. The motivation for the use of words or sentences instead of numbers is that linguistic classification and characterizations are generally less precise than numerical ones. In this research article, we encourage the fuzzy linguistic approach and introduce the novel concept known as m-polar fuzzy linguistic variable (mFLV) to increase the affluence of linguistic variables based on m-polar fuzzy (mF) approach. An mF set is an effective concept for interpreting uncertainty and fuzziness. The concept of mFLV is more versatile and sensible for dealing with real-life problems, when data comes from qualitative and multipolar information. We also introduce an mF linguistic ELECTRE-I approach to solve multiple-criteria decision-making (MCDM) and multiple-criteria group decision-making (MCGDM) problems, where the evaluation of the alternatives under suitable linguistic values are determined by the decision-makers. Furthermore, we validate the efficiency of our proposed technique by applying it to real-life examples, such as the salary analysis of companies and by selecting a corrupt country. Finally, we develop an algorithm of our proposed approach, present its flow chart, and generate computer programming code.

The energy of a simple connected graphGis equal to the sum of the absolute value of eigenvalues of the graphGwhere the eigenvalue of a graphGis the eigenvalue of its adjacency matrixAG. Ultimately, scores of various graph energies have been originated. It has been shown in this paper that the different graph energies of the regular splitting graphS′Gis a multiple of corresponding energy of a given graphG.

Let (V,0) be the germ of an analytic variety in and f an analytic function germ defined on V. For functions with isolated singularity on V, Bruce and Roberts introduced a generalization of the Milnor number of f, which we call Bruce–Roberts number, μBR(V,f). Like the Milnor number of f, this number shows some properties of f and V. In this paper we investigate algebraic and geometric characterizations of the constancy of the Bruce–Roberts number for families of functions with isolated singularities on V. We also discuss the topological invariance of the Bruce–Roberts number for families of quasihomogeneous functions defined on quasihomogeneous varieties. As application of the results, we prove a relative version of the Zariski multiplicity conjecture for quasihomogeneous varieties.

Let Δ _Γ ( G ) be Gallai simplicial complex defined as a generalization of Gallai graph Γ ( G ) such that G is a finite simple graph. The Betti numbers are topological objects which were proved to be invariants by Poincaré. In this note, we give formula for Betti numbers of Gallai simplicial complex associated to prism graph by coding in MATLAB. We prove that a simple graph G having n vertices of the form n = 3 l + 2 or 3 l + 3 is f -Gallai graph for G = ???? 4 l (respectively G = ???? 4 l + 1 ) a graph consisting of star graphs S _{2 l} and S 2 l ′ (respectively S _{2 l} and S _{2 l +1} ) with l common vertices such that l ≥ 2.

Silicon carbide (SiC), also called carborundum, is a semiconductor containing silicon and carbon. Dendrimers are repetitively branched molecules that are typically symmetric around the core and often adopt a spherical three-dimensional morphology. Bismuth(III) iodide is an inorganic compound with the formulaBiI3. This gray-black solid is the product of the reaction between bismuth and iodine, which once was of interest in qualitative inorganic analysis. In chemical graph theory, we associate a graph to a compound and compute topological indices that help us in guessing properties of the understudy compound. A topological index is the graph invariant number, calculated from a graph representing a molecule. Most of the proposed topological indices are related either to a vertex adjacency relationship (atom-atom connectivity) in the graph or to topological distances in the graph. In this paper, we aim to compute the first and second Gourava indices and hyper-Gourava indices for silicon carbides, bismuth(III) iodide, and dendrimers.

A more general class than complete intersection singularities is the class of determinantal singularities. They are defined by the vanishing of all the minors of a certain size of an \(m\times n\)-matrix. In this note, we consider \(\mathcal {G}\)-finite determinacy of matrices defining a special class of determinantal varieties. They are called essentially isolated determinantal singularities (EIDS) and were defined by Ebeling and Gusein-Zade (Singul Appl 267:119–131, 2009 ). In this note, we prove that matrices parametrized by generic homogeneous forms of degree d define EIDS. It follows that \(\mathcal {G}\)-finite determinacy of matrices holds in general. As a consequence, EIDS of a given type (m, n, t) holds in general.