Abstract
In many practical problems, randomness and uncertainty simultaneously appear in one complex system or network. When graph theory is applied to these problems, these complex systems or networks are usually represented by uncertain random graphs, in which some edges exist with degrees in probability measure, and some other edges exist with degrees in uncertain measure. In this paper, we focus on the connectivity of uncertain random graphs with respect to vertices. We propose the concepts of k -vertex-connectivity index and the connectivity of an uncertain random graph. The former is the chance measure that an uncertain random graph is k-vertex-connected, and the latter is an uncertain random variable, which characterizes the connectivity of the uncertain random graph with respect to vertices. We discuss some properties of these concepts. Methods and formulas are also presented for calculating the k -vertex-connectivity index, and the distribution and expected value of the connectivity of an uncertain random graph.
Highlights
The study of graph theory could be traced back to the year of 1736, when Euler published the first paper on graph theory for solving the Seven Bridges Problem of Königsberg
König published his book Theory of Finite and Infinite Graphs in 1936, which is the first book on graph theory. (König’s book was republished in 1990 [18].) Since this subject experienced explosive growth, due in large measure to its role as an essential structure underpinning modern applied mathematics
Graph theory has been widely applied to many fields, especially to all kinds of networks, such as social network [3], web network [4] and biological protein network [29]
Summary
The study of graph theory could be traced back to the year of 1736, when Euler published the first paper on graph theory for solving the Seven Bridges Problem of Königsberg. Euler used vertices to represent the areas of land, and used edges to represent the bridges This is the basic model of graphs of graph theory, and is still widely used today. When there was sufficient data to generate probability distribution functions for random variables, Erdös and Rényi suggested that whether the edge between two vertices exists or not, could be represented by a random variable Under this assumption, the model of random graphs was proposed by them [6]. In order to represent these systems and networks, uncertain random graph was proposed via chance theory by Liu [22] in 2014. We will discuss the connectivity properties of an uncertain random graph with respect to vertices. The last section will conclude this paper with a brief summary
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