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A graph containing some edges with probability measures and other edges with uncertain measures is referred to as an uncertain random graph. Numerous real-world problems in social networks and transportation networks can be boiled down to optimization problems in uncertain random graphs. Actually, information in optimization problems in uncertain random graphs is always asymmetric. Regularization is a common optimization problem in graph theory, and the regularity index is a fundamentally measurable indicator of graphs. Therefore, this paper investigates the regularity index of an uncertain random graph within the framework of chance theory and information asymmetry theory. The concepts of k-regularity index and regularity index of the uncertain random graph are first presented on the basis of the chance theory. Then, in order to compute the k-regularity index and the regularity index of the uncertain random graph, a simple and straightforward calculating approach is presented and discussed. Furthermore, we discuss the relationship between the regularity index and the k-regularity index of the uncertain random graph. Additionally, an adjacency matrix-based algorithm that can compute the k-regularity index of the uncertain random graph is provided. Some specific examples are given to illustrate the proposed method and algorithm. Finally, we conclude by highlighting some potential applications of uncertain random graphs in social networks and transportation networks, as well as the future vision of its combination with symmetry. [ABSTRACT FROM AUTHOR]

As the system becomes more and more complex, we are usually in the state of indeterminacy. In the real world, the states of uncertainty and randomness are the two most common types of indeterminacy. An uncertain random graph is applied to describe a graph model with uncertainty and randomness simultaneously. This paper mainly focuses on the connectivity of two vertices in an uncertain random graph. Firstly, a local connectivity index is proposed to unveil the chance measure that two special vertices are connected in an uncertain random graph. Furthermore, a method for calculating the local connectivity index is formulated. In addition, some simplified forms of the method are developed, and an algorithm is designed to obtain the local connectivity index. Finally, the information relevant to the relationship between the local connectivity index and the connectivity index is discussed.

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.

Connectivity is one of the most important concepts in graph theory. When graph theory is applied to complex systems with indeterminate factors, uncertainty and randomness are two basic types of indeterminacy. Therefore, the model of uncertain random graph was proposed via chance theory, in which some edges exist with degrees in probability measure and others exist with degrees in uncertain measure. This paper discusses the connectivity of an uncertain random graph with respect to edges. Concepts of k-connectivity index and edge-connectivity of an uncertain random graph are proposed. Methods and algorithms for calculating the k-connectivity index and the distribution of edge-connectivity are also proposed. In addition, some properties of them are discussed.

A graph containing some edges with probability measures and other edges with uncertain measures is referred to as an uncertain random graph. Numerous real-world problems in social networks and transportation networks can be boiled down to optimization problems in uncertain random graphs. Actually, information in optimization problems in uncertain random graphs is always asymmetric. Regularization is a common optimization problem in graph theory, and the regularity index is a fundamentally measurable indicator of graphs. Therefore, this paper investigates the regularity index of an uncertain random graph within the framework of chance theory and information asymmetry theory. The concepts of k-regularity index and regularity index of the uncertain random graph are first presented on the basis of the chance theory. Then, in order to compute the k-regularity index and the regularity index of the uncertain random graph, a simple and straightforward calculating approach is presented and discussed. Furthermore, we discuss the relationship between the regularity index and the k-regularity index of the uncertain random graph. Additionally, an adjacency matrix-based algorithm that can compute the k-regularity index of the uncertain random graph is provided. Some specific examples are given to illustrate the proposed method and algorithm. Finally, we conclude by highlighting some potential applications of uncertain random graphs in social networks and transportation networks, as well as the future vision of its combination with symmetry.

In practical applications of graph theory, indeterminacy factors always appear in graphs. Uncertain random graph was proposed via chance theory, in which some edges exist with degrees in probability measure and others exist with degrees in uncertain measure. This paper discusses the contributions of edges for connectivity of an uncertain random graph and proposes concepts about significance of edges, according to which edges are classified. In addition, this paper presents algorithms for calculating connectivity index and significance of edges of an uncertain random graph. Examples are given to illustrate algorithms and methods. [ABSTRACT FROM AUTHOR]

With the increasing of the complexity of a system, there is a variety of indeterminacy in the practical applications of graph theory. We focus on uncertain random graph, in which some edges exist with degrees in probability measure and others exist with degrees in uncertain measure. In this paper, the chance theory is applied to construct the cycle index of an uncertain random graph. Then a method to calculate the cycle index of an uncertain random graph is presented. We also discuss some properties of the cycle index. [ABSTRACT FROM AUTHOR]

We are usually in the state of indeterminacy. Uncertainty and randomness are two basic types of indeterminacy. In many cases, uncertainty and randomness exist simultaneously in a complex system. This paper considers the Euler tour of an uncertain random graph, in which some edges exist with some degrees in uncertain measure and others exist with some degrees in probability measure. In order to show how likely an uncertain random graph is Eulerian, an Euler index of an uncertain random graph is proposed first. Then a method to calculate the Euler index of an uncertain random graph is given. In addition, some properties of the Euler index are discussed. [ABSTRACT FROM AUTHOR]