Prediction of links/ connections in social networks is useful for increasing business in the area of telecom and social media. All types of social networking organization are trying to increase their nodes, i.e. the number of members. So the calculation of link prediction is the most important task because this calculation will help how to increase the number of user on a social network. A new technique of link prediction, namely RSM index, is introduced in this paper. Few essential properties have been established. Also, a comparative study with the existing methods is depicted with suitable tables and graphs.

Colouring of graphs is being used in several representations of real world systems like map colouring, traffic signalling, etc. This study introduces the edge colouring of fuzzy graphs. The chromatic index and the strong chromatic index are defined and related properties are investigated. In addition, job oriented web sites, traffic light problems have been presented and solved using the edge colouring of fuzzy graphs more effectively.

This paper presents a comprehensive study of the linguistic Z-graph, which is a novel framework designed to analyze linguistic structures within social networks. By integrating concepts from graph theory and linguistics, the linguistic Z-graph provides a detailed understanding of language dynamics in online communities. This study highlights the practical applications of linguistic Z-graphs in identifying central nodes within social networks, which are crucial for online businesses in market capture and information dissemination. Traditional methods for identifying central nodes rely on direct connections, but social network connections often exhibit uncertainty. This paper focuses on using fuzzy theory, particularly linguistic Z-graphs, to address this uncertainty, offering more detailed insights compared to fuzzy graphs. Our study introduces a new centrality measure using linguistic Z-graphs, enhancing our understanding of social network structures.