This work introduces the notion of a hesitant bipolar-valued intuitionistic fuzzy graph (HBVIFG), which reflects four different characterizations: membership with positive/negative aspects and non-membership with positive/negative aspects, incorporating multi-dimensional alternatives in all of its information. HBVIFG generalizes both HBVFG and BVHFG due to its diversified nature in observing four perspectives along with multiple attributes in a piece of information. Numerous studies, examples, and graphical representations emphasize the concept’s distinctiveness and importance. The following graph theory terms are defined: strong directed HBVIFG, full directed HBVIFG, directed spanning HBVIFSG, directed HBVIFSG, and partial directed hesitant bipolar-valued intuitionistic fuzzy subgraph (HBVIFSG). Examples of operations utilizing two HBVIFGs are Cartesian, direct, lexicographical, and strong products. A scenario is used to generate the mapping of relations, which includes homomorphism, isomorphism, weak isomorphism, and co-weak isomorphism. We describe a directed HBVIFG application that employs an algorithm to determine the most dominant person and self-persistent person in a social system and a comparative study is also provided. The proposed method provides a more detailed framework for assessing the most dominant and self-persistent individual in a social network across multi-level attributes along with positive and negative side membership and non-membership grades in each element of a network.

The fuzzy‐directed graph is an efficient tool to deal with the directional relationships among the nodes that possess imprecise and uncertain information. To handle the membership value of the edges that provide greater effect when two nodes are combined, we introduce inverse fuzzy‐directed graph (IFDG) . In this article, we introduce various types of subgraphs of IFDG, the degree of IFDG, and the underlying graph of IFDG. In addition, various IFDG operations, quasidirected union, quasidirected cartesian product, and converse of IFDG are defined and discussed in detail. Furthermore, the notion of isomorphism is defined for . Vertex connectedness of IFDG and its various forms, strongly connected, partially semiconnected, and weakly connected are introduced for and their related results are discussed briefly. Also, the necessary and sufficient conditions for strongly connected, partially semiconnected, and weakly connected IFDGS have been proven. The concluding segment demonstrates the practicality of employing the IFDG algorithm to identify the most congested traffic flows in Kumbakonam, Tamil Nadu, India.