Search

In this write-up, we explore the family of two-bridge knots for fault-tolerance. We findout fault-tolerant unknotting numbers for certain subfamilies of two-bridge knots. Besidesthis, we work out on special subfamilies C(a, a1, a2, a3, . . . ,±2,−ak,−ak−1, . . . ,−a2,−a1)and C(2n, 3) of two-bridge knots with unknotting number 1 and n respectively and found avarient of fault-tolerant unknotting number called restricetd fault-tolerant unknotting number.Lastly, this paper is concluded by explicating biological disposition of fault-tolerantunknotting number in terms of enzyme action on DNA.

The main purpose is to check the impact of social capital on households’ objective well-being in the context of a developing country like Pakistan. The primary data is collected from 250 households of eight towns in district Faisalabad, the third-largest most populous city of Pakistan. Ordered probit models are used for analysis. The results show that objective well-being is positively and significantly affected by social capital except for social participation and neighborhood cohesion. For subjective well-being, life satisfaction and well-being are positively and significantly associated with all dimensions of social capital except the perception of trust and safety, and education. In the case of self-perceived health, civic participation, voting behavior, and collecting actions (CVC), reciprocity and social support are insignificant. Our results may help in devising social policies related to improvement in life quality and well-being of people. It gives insight into the role of social capital for the reduction of poverty, ameliorating individuals' health status and the well-being of the people. It is suggested that society's well-being may be achieved with higher social capital. The government may devise strategies to increase social capital for improving the well-being of the targeted society.

Vukičević and Gasperov introduced the concept of 148 discrete Adriatic indices in 2010. These indices showed good predictive properties against the testing sets of the International Academy of Mathematical Chemistry. Among these indices, twenty indices were taken as beneficial predictors of physicochemical properties. The inverse sum indeg index denoted by ISI G k of G k is a notable predictor of total surface area for octane isomers and is presented as ISI G k = ∑ g k g k ′ ∈ E G k d G k g k d G k g k ′ / d G k g k + d G k g k ′ , where d G k g k represents the degree of g k ∈ V G k . In this paper, we determine sharp bounds for ISI index of graph operations, including the Cartesian product, tensor product, strong product, composition, disjunction, symmetric difference, corona product, Indu–Bala product, union of graphs, double graph, and strong double graph.

A vast amount of information about distance based graph invariants is contained in the Hosoya polynomial. Such an information is helpful to determine well-known distance based molecular descriptors. The Hosoya index orZ-index of a graphGis the total number of its matching. The Hosoya index is a prominent example of topological indices, which are of great interest in combinatorial chemistry, and later on it applies to address several chemical properties in molecular structures. In this article, we investigate Hosoya properties (Hosoya polynomial, reciprocal Hosoya polynomial and Hosoya index) of the commuting graph associated with an algebraic structure developed by the symmetries of regular molecular gones (constructed by atoms with regular atomic-bonding).

There are several approaches to lower the complexity of huge networks. One of the key notions is that of twin nodes, exhibiting the same connection pattern to the rest of the network. We extend this idea by defining a twin preserving spanning subgraph (TPS-subgraph) of a simple graph as a tool to compute certain graph related invariants which are preserved by the subgraph. We discuss how these subgraphs preserve some distance based parameters of the simple graph. We introduce a sub-skeleton graph on a vector space and examine its basic properties. The sub-skeleton graph is a TPS-subgraph of the non-zero component graph defined over a vector space. We prove that some parameters like the metric-dimension are preserved by the sub-skeleton graph.

The study of networks and graphs through structural properties is a massive area of research with developing significance. One of the methods used in studying structural properties is obtaining quantitative measures that encode structural data of the whole network by the real number. A large collection of numerical descriptors and associated graphs have been used to examine the whole structure of networks. In these analyses, degree-related topological indices have a significant position in theoretical chemistry and nanotechnology. Thus, the computation of degree-related indices is one of the successful topics of research. The general sum-connectivity GSC index of graph Q is described as χ ρ Q = ∑ qq ′ ∈ E Q d q + d q ′ ρ , where d q presents the degree of the vertex q in Q and ρ is a real number. The total graph T Q is a graph whose vertex set is V Q ∪ E Q , and two vertices are linked in T Q if and only if they are either adjacent or incident in Q . In this article, we study the general sum-connectivity index χ ρ Q of total graphs for different values of ρ by using Jensen’s inequality.

Starch is a polymer of glucose where alpha-linkages are associated with glucopyranose units. It comprises a mixture of amylose and amylopectin. Furthermore, amylose is a linear chain of hundreds of glucose molecules. Starches are not allowed to be dissolved in water. They can be digested by breaking down alpha bonds (glycosidic bonds). Its cyclic degradation products, called cyclodextrins, are the best role models for amylose. They can be considered simple turns of the amylose propeller that has imploded into a circular path. Both humans and animals have amylases, which allow them to digest starches. The important sources of starch include potatoes, rice, wheat, and maize for human consumption. The production of starches is how plants store glucose. The blue colour of starch produced by an iodine solution or iodine reaction is used for its identification. Polysaccharides with a reduced degree of polymerization, known as dextrins, are produced in the starch’s partial acid hydrolysis. Complete hydrolysis leads to glucose. In this article, we compute the topological properties: Zagreb index M 1 Γ and M 2 Γ , Randić index R α Γ for α = − 1 / 2 , − 1 , 1 / 2 , 1 , atom-bond connectivity index ABC Γ , geometric-arithmetic index GA Γ , fourth atom-bond connectivity index ABC 4 Γ , fifth geometric-arithmetic index GA 5 Γ , and degree-based topological indices of a graph Γ representing polysaccharides, namely, amylose and blue starch-iodine complex. In the end, we compare these indices and depict their graphic behavior.

In this research work, generalized thermal and mass transports for the unsteady flow model of an incompressible differential type fluid are considered. The Caputo–Fabrizio fractional derivative is used for the respective generalization of Fourier’s and Fick’s laws. A MHD fluid flow is considered near a flat vertical surface subject to unsteady mechanical, thermal, and mass conditions at boundary. The governing equations of flow model are solved by integral transform, and closed form results for generalized momentum, thermal, and concentration fields are obtained. Generalized thermal and mass fluxes at boundary are quantified in terms of Nusselt and Sherwood numbers, respectively, and presented in tabular form. The significance of the physical parameters over the momentum, thermal, and concentration profiles is characterized by sketching the graphs.

Let A be a domain. We relate the perinormality (as de ned by Epstein and Shapiro) of A and A[X] for a narrow class of Noetherian domains.

We extend to rings with zero-divisors the concept of perinormal domain introduced by N. Epstein and J. Shapiro. A ring $A$ is called perinormal if every overring of $A$ which satisfies going down over $A$ is $A$-flat. The Pr\"ufer rings and the Marot Krull rings are perinormal., Comment: 9 pages