Journal of Prime Research in Mathematics

Abstract: In this work, a fractional optimal control of corruption model is investigated. The variable controls are included in the model to optimize the best strategy in reducing the corruption in the society. The fraction derivative employed in the study is in Atangana–Beleanu–Caputo (ABC) sense based on generalized Mittag–Leffler. The uniqueness and existence of solution of the corruption model is established. The necessary and sufficient condition for establishing fractional optimal control in ABC sense is determined. A numerical algorithm for obtaining fractional optimal control solution is presented. The numerical solution results show that the best strategy in controlling corruption in the society is to optimize all the thee controls simultaneously.

Abstract: For a commutative semiring \(R\) with non-zero identity, the maximal graph of \(R\), denoted by \(MG(R)\), is the graph whose vertices are all elements of \(UM(R)\) with two distinct vertices joined by an edge when there is a maximal co-ideal that contains both of them. In this paper, we study some properties of maximal graph such as planarity, radius, splitting and domination number.

Abstract: In this article, we introduce a new class of generalized convex functions involving two arbitrary auxiliary functions \(h_1, h_2 : I = [a, b] ⊆ R → R\), which is called generalized \((h_1, h_2)\) convex functions. We obtain several new classes of convex functions as special cases. We derive some new integral inequalities for generalized convex functions. Several special cases are also discussed. Results proved in this paper can be viewed as new significant contributions in this area.

Abstract: Liu et al. [IEEE Access; 7(2019); 105479-105488] defined the concept of the generalized subdivided operations on graphs and obtained the generalized F-sum graphs. They also calculated the 1st and 2nd Zagreb indices of the generalized F-sum graphs. In the continuation of this work, we study the forgotten index (F-index) of the generalized F-sum graphs in terms of different topological indices (TI’s) of their base graphs. In the end, the results of F-index on the generalized F-sum graphs acquired by the particular classes of alkane are also included.

Abstract: A sum divisor cordial labeling of a graph \(G\) with vertex set \(V\) is a bijection \(f\) from \(V\) to \(\{1, 2, ··· , |V (G)|\}\) such that an edge \(uv\) is assigned the label 1 if 2 divides \(f(u) + f(v)\) and \(0\) otherwise; and the number of edges labeled with \(0\) and the number of edges labeled with 1 differ by at most 1. A graph with a sum divisor cordial labeling is called a sum divisor cordial graph. In this paper, we prove that \(P^{2}_{n}, P_n ⊙ mK_1, S(P_n ⊙ mK_1), D_2(P_n), T(P_n)\), the graph obtained by duplication of each vertex of path by an edge, \(T(C_n), D_2(C_n)\), the graph obtained by duplication of each vertex of cycle by an edge, C^{(t)}_{4}, book, quadrilateral snake and alternate triangular snake are sum divisor cordial graphs.

Abstract: The concept of picture hesitant fuzzy set (PHFS) is a generalization of picture fuzzy set (PFS) and hesitant fuzzy set (HFS). Such kind of idea can be very helpful in problems where opinions are of more than two types i.e. yes, no, abstinence and refusal. The goal of this manuscript is to introduce the concept of similarity measures for PHFSs as a generalization of the similarity measures for PFS. We studied the basic concepts of  PFSs, HFSs and PHFSs. We proposed some similarity measures for PHFSs such as cosine similarity measure, set-theoretic similarity measure and grey similarity measure for PHFSs. Some weighted similarity measures are also proposed where weight of the attributes is considered. Then these similarity measures for PHFSs are applied to a building material recognition problem. Finally, a comparative study of similarity measures of PHFSs is established with similarity measures of PFSs, HFSs, IFSs and IHFSs and the advantages of new work are studied.