We develop a new framework for analysing finite connected, oriented graphs of valency 4, which admit a vertex-transitive and edge-transitive group of automorphisms preserving the edge orientation. We identify a subfamily of 'basic' graphs such that each graph of this type is a normal cover of at least one basic graph. The basic graphs either admit an edge-transitive group of automorphisms that is quasiprimitive or biquasiprimitive on vertices, or admit an (oriented or unoriented) cycle as a normal quotient. We anticipate that each of these additional properties will facilitate effective further analysis, and we demonstrate that this is so for the quasiprimitive basic graphs. Here we obtain strong restirictions on the group involved, and construct several infinite families of such graphs which, to our knowledge, are different from any recorded in the literature so far. Several open problems are posed in the paper.

In fuzzy set theory, t-norms and t-conorms are fundamental binary operators. Yager proposed respective parametric families of both t-norms and t-conorms. In this paper, we apply these operators for the analysis of Pythagorean fuzzy sets. For this purpose, we introduce six families of aggregation operators named Pythagorean fuzzy Yager weighted averaging aggregation, Pythagorean fuzzy Yager ordered weighted averaging aggregation, Pythagorean fuzzy Yager hybrid weighted averaging aggregation, Pythagorean fuzzy Yager weighted geometric aggregation, Pythagorean fuzzy Yager ordered weighted geometric aggregation and Pythagorean fuzzy Yager hybrid weighted geometric aggregation. These tools inherit the operational advantages of the Yager parametric families. They enable us to study two multi-attribute decision-making problems. Ultimately we can choose the best option by comparison of the aggregate outputs through score values. We show this procedure with two practical fully developed examples.

Graph Theory International audience Ruskey and Savage conjectured that in the d-dimensional hypercube, every matching M can be extended to a Hamiltonian cycle. Fink verified this for every perfect matching M, remarkably even if M contains external edges. We prove that this property also holds for sparse spanning regular subgraphs of the cubes: for every d ≥7 and every k, where 7 ≤k ≤d, the d-dimensional hypercube contains a k-regular spanning subgraph such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle. We do not know if this result can be extended to k=4,5,6. It cannot be extended to k=3. Indeed, there are only three 3-regular graphs such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle, namely the complete graph on 4 vertices, the complete bipartite 3-regular graph on 6 vertices and the 3-cube on 8 vertices. Also, we do not know if there are graphs of girth at least 5 with this matching-extendability property.

Abstract. We study finite four-valent graphs Γ admitting an edge-transitive group G of automorphisms such that G determines and preserves an edgeorientation on Γ, and such that at least one G-normal quotient is a cycle (a quotient modulo the orbits of a normal subgroup of G). We show on the one hand that the number of distinct cyclic G-normal quotients can be unboundedly large. On the other hand existence of independent cyclic G-normal quotients (that is, they are not extendable to a common cyclic G-normal quotient) places severe restrictions on the graph Γ and we classify all examples. We show there are five infinite families of such pairs (Γ, G), and in particular that all such graphs have at least one normal quotient which is an unoriented cycle. We compare this new approach with existing treatments for the subclass of weak metacirculant graphs with these properties, finding that only two infinite families of examples occur in common from both analyses. Several open problems are posed.

We analyse the normal quotient structure of several infinite families of finite connected edge-transitive, four-valent oriented graphs. These families were singled out by Marušič and others to illustrate various different internal structures for these graphs in terms of their alternating cycles (cycles in which consecutive edges have opposite orientations). Studying the normal quotients gives fresh insights into these oriented graphs: in particular we discovered some unexpected 'cross-overs' between these graph families when we formed normal quotients. We determine which of these oriented graphs are 'basic', in the sense that their only proper normal quotients are degenerate. Moreover, we show that the three types of edge-orientations studied are the only orientations, of the underlying undirected graphs in these families, which are invariant under a group action which is both vertex-transitive and edge-transitive. CC This work is licensed under http://creativecommons.org/licenses/by/3.0/

Despite the importance of divergence measures, the literature has not provided a satisfactory formulation for the case of q-rung orthopair fuzzy set. This paper criticizes the existing attempts in terms of respect of the basic axioms of a divergence measure. Then new improved, axiomatically supported divergence measures for qROFSs are proposed. Additional properties of the new divergence measures are discussed to guarantee their good performance. The transformation relationships with entropy and dissimilarity measures are debated. The multiattribute border approximation area comparison decision method is extended based on the suggested divergence measures, and it is applied to the selection of all-rounder cricketer for a team.

This paper presents the novel concept of complex spherical fuzzy N -soft set ( C S F N S f S ) which is capable of handling two-dimensional vague information with parameterized ranking systems. First, we propose the basic notions for a theoretical development of C S F N S f S s , including ranking functions, comparison rule, and fundamental operations (complement, union, intersection, sum, and product). Furthermore, we look into some properties of C S F N S f S s . We then produce three algorithms for multiattribute decision-making that take advantage of these elements. We demonstrate their applicability with the assistance of a numerical problem (selection of best third-party app of the year). A comparison with the performance of Pythagorean N -soft sets speaks for the superiority of our approach. Moreover, with an aim to expand the range of techniques for multiattribute group decision-making problems, we design a C S F N S f -TOPSIS method. We use a complex spherical fuzzy N -soft weighted average operator in order to aggregate the decisions of all experts according to the power of the attributes and features of alternatives. We present normalized-Euclidean distances (from the alternatives to both the C S F N S f positive and negative ideal solutions, respectively) and revised closeness index in order to produce a best feasible alternative. As an illustration, we design a mathematical model for the selection of the best physiotherapist doctor of Mayo hospital, Lahore. We conduct a comparison with the existing complex spherical fuzzy TOPSIS method that confirms the stability of the proposed model and the reliability of its results.

The preference ranking organization method for enrichment of evaluations (PROMETHEE) method considers a significant outranking class of multi-criteria decision analysis (MCDA), as it is easy to deal with its simple computations. In the PROMETHEE, different preference functions are used according to the type and nature of attributes or criteria that demonstrate the clearness and reliability of this method. This study provides a new version of the PROMETHEE method using bipolar fuzzy information, named the bipolar fuzzy PROMETHEE method. Bipolar fuzzy sets or numbers constitute an asymmetrical relationship between two judgmental factors of human reasoning. Vague and imprecise knowledge is characterized by bipolar fuzzy linguistic terms which are further represented in the form of trapezoidal bipolar fuzzy numbers. The trapezoidal bipolar fuzzy numbers are used by analysts to assign the preferences of alternatives on the basis of criteria. Further, a ranking function of bipolar fuzzy numbers is considered to access the crisp real preferences of alternatives. The entropy weighting information is employed to calculate the weights of attributes by considering the condition of normality. A numerical example such as the selection of green suppliers by using the bipolar fuzzy PROMETHEE is performed on the basis of the usual criterion preference function in order to explain the procedure of the proposed method. Comparable results are derived by using the combination of linear and level preference functions. The results obtained by using different types of preference functions are the same, representing the authenticity of the proposed bipolar fuzzy PROMETHEE method.