Neutrosophic graphs are used to model inconsistent information and imprecise data about any real-life problem. It is regarded as a generalization of intuitionistic fuzzy graphs. Since interval-valued neutrosophic sets are more accurate, compatible, and flexible than single neutrosophic sets, interval-valued neutrosophic graphs (IVNGs) were defined. The interval-valued neutrosophic graph is a fundamental issue in graph theory that has wide applications in the real world. Also, problems may arise when partial ignorance exists in the datasets of membership [0, 1], and then, the concept of IVNG is crucial to represent the problems. Line graphs of neutrosophic graphs are significant due to their ability to represent and analyze uncertain or indeterminate information about edge relationships and complex networks in graphs. However, there is a research gap on the line graph of interval-valued neutrosophic graphs. In this paper, we introduce the theory of an interval-valued neutrosophic line graph (IVNLG) and its application. In line with that, some mathematical properties such as weak vertex isomorphism, weak edge isomorphism, effective edge, and other properties of IVNLGs are proposed. In addition, we defined the vertex degree of IVNLG with some properties, and by presenting several theorems and propositions, the relationship between fuzzy graph extensions and IVNLGs was explored. Finally, an overview of the algorithm used to solve the problems and the practical application of the introduced graphs were provided.

中文翻译：

---

区间值中智线图的理论与应用

中智图用于对有关任何现实生活问题的不一致信息和不精确数据进行建模。它被认为是直觉模糊图的推广。由于区间值中智集比单个中智集更准确、更兼容、更灵活，因此定义了区间值中智图 (IVNG)。区间值中智图是图论中的一个基本问题，在现实世界中有着广泛的应用。此外，当隶属度数据集 [0, 1] 中存在部分无知时，可能会出现问题，此时 IVNG 的概念对于表示问题至关重要。中智图的线图非常重要，因为它们能够表示和分析有关图中边缘关系和复杂网络的不确定或不确定信息。然而，区间值中智图的线图还存在研究空白。在本文中，我们介绍了区间值中智线图（IVNLG）的理论及其应用。据此，提出了IVNLG的弱顶点同构、弱边同构、有效边等数学性质。此外，我们定义了IVNLG的顶点度及其一些性质，并通过提出几个定理和命题，探讨了模糊图扩展与IVNLG之间的关系。最后，概述了用于解决问题的算法以及所介绍的图的实际应用。