Tripolar fuzzy sets offer a robust framework that integrates the aspects of intuitionistic fuzzy sets and bipolar fuzzy sets. Complex decision-making problems often involve multiple attributes with specific properties that reflect both negative and positive aspects. In such cases, intuitionistic fuzzy sets (IFS) and bipolar fuzzy sets (BFS) may be insufficient to handle these complexities that makes tripolar fuzzy sets an essential tool for identifying optimal solutions. In our study, we introduce a Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method using connection number of set pair analysis under the tripolar fuzzy environment. Initially, we express tripolar fuzzy number as a connection number, which is a pivotal concept in set pair analysis. We then extend the TOPSIS method based on the connection number of set pair analysis within the realm of tripolar fuzzy sets. By applying the proposed methodology, we provide the computational framework for the problem related to multi-criteria decision making. Moreover, we observe that our proposed technique also satisfies the existing test criterions. In this context, we conduct the comparative study of the newly established technique with the other ones existing in intuitionistic fuzzy and bipolar fuzzy environments. Finally, we show that our proposed technique is superior than the others.

Connectivity in graphs is useful in describing different types of communication systems like neural networks, computer networks, etc. In the design of any network, it is essential to evaluate the connections based on their strengths. In this manuscript, we comprehensively describe various connectivity parameters related to interval-valued intuitionistic fuzzy graphs (IVIFGs). These are the generalizations of the parameters defined for fuzzy graphs, interval-valued fuzzy graphs, and intuitionistic fuzzy graphs. First, we introduce interval-valued intuitionistic fuzzy bridges (IVIF bridges) and interval-valued intuitionistic fuzzy cut-nodes (IVIF cut-nodes). We discuss the many characteristics of these terms as well as establish the necessary and sufficient conditions for an arc to become an IVIF-bridge and a vertex to be an IVIF-cutnode. Furthermore, we initiate the concepts of interval-valued intuitionistic fuzzy cycles (IVIFCs) and interval-valued intuitionistic fuzzy trees (IVIFTs) and explore few relationships among them. In addition, we introduce the notions of fuzzy blocks and fuzzy block graphs and extend these terms as interval-valued fuzzy blocks (IVF-blocks) and interval-valued intuitionistic fuzzy block graphs (IVIF-block graphs). Finally, we provide the application of interval-valued intuitionistic fuzzy trees (IVIFTs) in a road transport network.