Closeness and Residual Closeness of Harary Graphs
Hande Tuncel Golpek, Aysun Aytac
TL;DR
This work investigates vulnerability of Harary graphs through closeness and vertex residual closeness. It develops exact closed-form expressions for the closeness of Harary graphs $H_{k,n}$ under various parity and diameter regimes, and links Harary graphs to Consecutive Circulant Graphs to facilitate computations. It then derives explicit formulas for vertex residual closeness $R$ after removal of a vertex, across multiple parity, modular, and diameter cases, highlighting which vertices are most critical. The results advance the theoretical understanding of network robustness in highly connected, edge-minimal graphs and provide precise metrics for assessing and designing resilient interconnection structures.
Abstract
Analysis of a network in terms of vulnerability is one of the most significant problems. Graph theory serves as a valuable tool for solving complex network problems, and there exist numerous graph-theoretic parameters to analyze the system's stability. Among these parameters, the closeness parameter stands out as one of the most commonly used vulnerability metrics. Its definition has evolved to enhance the ease of formulation and applicability to disconnected structures. Furthermore, based on the closeness parameter, vertex residual closeness, which is a newer and more sensitive parameter compared to other existing parameters, has been introduced as a new graph vulnerability index by Dangalchev. In this study, the outcomes of the closeness and vertex residual closeness parameters in Harary Graphs have been examined. Harary Graphs are well-known constructs that are distinguished by having $n$ vertices that are $k$-connected with the least possible number of edges.