Wiener index and vulnerability parameters of graphs (Q6166041)

The Wiener index of a finite simple graph \(G\), is defined as the sum of the distance between all pairs of vertices in \(G\). This index was originally proposed by \textit{H. Wiener} [J. Am. Chem. Soc. 69, No. 1, 17--20 (1947; \url{doi.org/10.1021/ja01193a005})] for analyzing the structural graphs of molecules. Over the years, various theoretical results have been derived regarding the Wiener index. The authors of this paper focus on the graph parameters integrity, toughness, tenacity, and binding number. These parameters are important in determining properties of graphs. This paper states that lower bounds on these parameters have been extensively used for this purpose. The authors contribute to the existing body of knowledge by providing the best possible upper bounds on the Wiener index of \(G\). These upper bounds guarantee a specified lower bound on each of the four parameters mentioned earlier. Overall, the paper highlights the significance of the Wiener index in graph analysis, particularly in relation to integrity, toughness, tenacity, and binding number. The authors' findings provide valuable insights into the relationship between the Wiener index and these graph parameters.