Closeness and Residual Closeness of Harary Graphs
For graph theorists and network analysts, this work offers closed-form expressions for two vulnerability parameters on a specific class of graphs, but it is an incremental contribution as it applies existing definitions to a known graph family.
This paper computes the closeness and vertex residual closeness parameters for Harary graphs, which are k-connected graphs with minimal edges. The results provide exact formulas for these vulnerability metrics.
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.