🤖 AI Summary
This study investigates the structural robustness of networks under node failures, with a focus on closeness centrality and its residual counterpart. By conducting the first systematic analysis of closeness and residual closeness in intermediate graphs—leveraging graph theory, line graph theory, and algorithmic design—it establishes precise relationships among the original graph, its line graph, and associated intermediate graphs. The main contributions include deriving exact closed-form expressions for closeness in intermediate graphs of several special graph classes, establishing general upper and lower bounds for residual closeness across broader graph families, and proposing an efficient algorithm for computing these measures. Experimental results demonstrate the superior performance of the proposed method, validating both its theoretical soundness and practical efficacy.
📝 Abstract
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.