Paulraj Gnanachandra

We introduce a neutrosophic framework for studying randomized rounding methods for the Traveling Salesman Problem (TSP). The approach is motivated by the work of Gharan, Saberi, and Singh on randomized rounding for the graphic TSP. In the proposed model, each edge is endowed with a cost interval and a neutrosophic assessment consisting of truth, indeterminacy, and falsity degrees. A scalarization rule converts this information into an effective edge cost, allowing classical linear programming relaxations and combinatorial optimization methods to be applied. We formulate a neutrosophic subtour relaxation, discuss randomized spanning-tree selection, and describe the role of minimum-cost T-joins in constructing tours. We also establish elementary properties of the scalarized model and give a conditional approximation result. The conditions needed for such a guarantee are stated explicitly; in particular, the classical approximation bound does not follow from neutrosophic notation alone. The paper provides a mathematical starting point for incorporating uncertain, incomplete, and conflicting edge-cost information into randomized approximation algorithms for the TSP.

Throughout this article, we propose an extension of the metric space due to M. Jleli and B. Samet (JS-metric space). This extension is made using the altering distance function, hence the name altering JS-metric. The idea of fixed points for the contraction mappings endowed with graph was first introduced by J. Jachymski in 2008. After that many researchers followed this idea and proposed various contraction mappings endowed with a graph. Motivated by these ideas, we have generalised some well-known contraction mappings in the literature that are endowed with a graph. Additionally, we have provided some fixed point results concerning these contractions in an altering JS-metric space.

The paper aims at initiating and exploring the concept of extended metric known as the Strong Altering JS-metric, a stronger version of the Altering JS-metric. The interrelation of Strong Altering JS-metric with the b-metric and dislocated metric has been analyzed and some examples have been provided. Certain theorems on fixed points for expansive self-mappings in the setting of complete Strong Altering JS-metric space have also been discussed.

Relation on a set is a simple mathematical model to which many real-life data can be connected. A binary relation on a set can always be represented by a digraph. Topology on a set can be generated by binary relations on the set . In this direction, the study will consider different classical categories of topological spaces whose topology is defined by the binary relations adjacency and reachability on the vertex set of a directed graph. This paper analyses some properties of these topologies and studies the properties of closure and interior of the vertex set of subgraphs of a digraph. Further, some applications of topology generated by digraphs in the study of biological systems are cited.