Solution to an open problem on the closeness of graphs

A network can be analyzed by means of many graph theoretical parameters. In the context of networks analysis, closeness is a structural metric that evaluates a node's significance inside a network. A cactus is a connected graph in which any block is either a cut edge or a cycle. This paper analyzes the closeness of cacti, we determine the unique graph that minimizes the closeness over all cacti with fixed numbers of vertices and cycles, which solves an open problem proposed by Poklukar \& \v{Z}erovnik [Fundam. Inform. 167 (2019) 219--234]

Closeness and Residual Closeness of Harary Graphs

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 metric. Its definition has evolved over time to enhance ease of formulation and applicability to disconnected structures. Furthermore, based on the closeness parameter, 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 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.Comment: 21 pages preprin