Analyzing Network Robustness via Residual Closeness
For graph theorists and network robustness researchers, this provides theoretical results and an algorithm for analyzing vertex failure resilience in middle graphs, though the contribution is incremental.
This paper derives exact expressions for closeness and residual closeness in middle graphs of certain graph classes, establishes general bounds, and proposes an algorithm for computing closeness in middle graphs with performance analysis.
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.