Networks are inherently vulnerable to vertex failures, making the analysis of their structural robustness a fundamental problem in graph theory. In this study, we investigate the closeness and vertex residual closeness of graphs, with a particular focus on the middle graph representations of certain special graph classes, which provide a richer structural framework for analysis. We derive exact expressions for the closeness values of these middle graphs and determine their residual closeness under vertex failures. By utilizing results obtained from specific graph families, we establish several general bounds for broader graph classes. Furthermore, by exploiting the relationship between the closeness of a graph, its line graphs, and middle graphs, we obtain new results that relate these three structures. In addition, we propose an algorithm for computing closeness in middle graphs and provide a detailed analysis of its performance.

翻译：网络本质上易受顶点失效的影响，这使得其结构鲁棒性分析成为图论中的基础问题。本研究探究了图的接近度与顶点残留接近度，重点聚焦于特定特殊图类的中图表示，该表示为分析提供了更丰富的结构框架。我们推导了这些中图接近度的精确表达式，并确定了其在顶点失效条件下的残留接近度。通过利用特定图族的结果，我们建立了更广泛图类的若干一般性界。此外，利用图的接近度、线图与中图之间的关系，我们获得了关联这三种结构的新结论。我们还提出了一种计算中图接近度的算法，并对其性能进行了详细分析。