Closeness and Residual Closeness of Harary Graphs

TL;DR

This paper investigates the properties of closeness and residual closeness parameters in Harary Graphs, which are highly connected minimal-edge graphs, to understand their implications for network vulnerability analysis.

Contribution

It provides a novel analysis of how closeness and residual closeness behave specifically in Harary Graphs, expanding understanding of their vulnerability metrics.

Findings

Analyzed closeness and residual closeness in Harary Graphs.

Identified how these parameters relate to network vulnerability.

Enhanced understanding of graph robustness metrics.

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.