This paper introduces the concept of hesitant bipolar-valued fuzzy graph (HBVFG), which captures the two opposing perspectives, namely the positive and negative opinions. The novelty, importance and implications of this concept are illustrated by some results, examples, and graphical representations. There are, respectively, some theoretical terms of graphs such as partial directed hesitant bipolar-valued fuzzy subgraph (HBVFSG), directed HBVFSG, directed spanning HBVFSG, strong directed HBVFG and complete directed HBVFG which are introduced. The operations, such as Cartesian, direct, lexicographical, and strong products, are also defined between two HBVFGs with examples. The mapping relations, such as homomorphism, isomorphism, weak isomorphism, and co-weak isomorphism, are derived with an example. The applications of directed HBVFGs with algorithms for finding the optimal path in a network and the dominant node and influence of index with the self-persistence degree of a node in a social network are presented. For each problem, an algorithm is developed and its effectiveness is demonstrated by examples. The proposed concept is assessed in terms of theory and practice. The benefits of the proposed solution are highlighted and a clear comparison is made with the existing methods.

本文引入了犹豫双极值模糊图（HBVFG）的概念，它捕获了两种相反的观点，即正面观点和负面观点。一些结果、例子和图形表示说明了这个概念的新颖性、重要性和含义。分别介绍了部分有向犹豫双极值模糊子图(HBVFSG)、有向HBVFSG、有向跨越HBVFSG、强有向HBVFG和完全有向HBVFG等图的理论术语。还通过示例在两个 HBVFG 之间定义了笛卡尔、直接、字典和强乘积等运算。并举例推导了同态、同构、弱同构、共弱同构等映射关系。介绍了定向 HBVFG 的应用，该算法用于寻找网络中的最佳路径和主导节点，以及社交网络中节点的自持续度对指标的影响。针对每个问题，开发了一种算法，并通过示例证明了其有效性。所提出的概念从理论和实践方面进行了评估。强调了所提出的解决方案的优点，并与现有方法进行了清晰的比较。