Abstract
Complex networks are ubiquitous in biological, physical and social sciences. Network robustness research aims at finding a measure to quantify network robustness. A number of Wiener type indices have recently been incorporated as distance-based descriptors of complex networks. Wiener type indices are known to depend both on the network’s number of nodes and topology. The Wiener polarity index is also related to the cluster coefficient of networks. In this paper, based on some graph transformations, we determine the sharp upper bound of the Wiener polarity index among all bicyclic networks. These bounds help to understand the underlying quantitative graph measures in depth.
Highlights
Complex networks are ubiquitous in biological, physical and social sciences
We provide a formula of the Wiener polarity index of bicyclic networks, from which the value of the index can be computed
We introduce three graph transformations, which can be used to increase the values of Wiener polarity index
Summary
Complex networks are ubiquitous in biological, physical and social sciences. Network robustness research aims at finding a measure to quantify network robustness. Quite a lot of different approaches to capture the robustness properties of a network have been undertaken. All of these approached are based on the analysis of the underlying graph—consisting of a set of vertices connected by edges of a network[1,2,3,4,5,6]. The use of Wiener index and related type of indices dates back to the seminal work of Wiener in 19477 Wiener introduced his celebrated index to predict the physical properties, such as boiling point, heats of isomerization and differences in heats of vaporization, of isomers of paraffin by their chemical structures. The extremal Wiener polarity index of (chemical) trees with given www.nature.com/scientificreports/
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