Matching Index of Uncertain Graph: Concept and Algorithm

Matching Index of Uncertain Graph: Concept and Algorithm Bo Zhang1 , Jin Peng2,∗ 1

School of Mathematics and Statistics, Huazhong Normal University Hubei 430079, China 2

Institute of Uncertain Systems, Huanggang Normal University Hubei 438000, China

Abstract: In practical applications of graph theory, there is no doubt that some uncertain factors may appear in graphs. This paper employs the uncertainty theory to deal with uncertain factors in uncertain graph. Matching index and perfect matching index of uncertain graph are proposed. Some properties of the matching index are discussed. Furthermore, we give an algorithm to calculate the matching index of uncertain graph. Keywords: Uncertainty theory; Uncertain measure; Matching index; Perfect matching index; Uncertain graph

1

Introduction

Since originated in seven bridges problem by Euler in 1736, the graph theory has received great attention. For instance, Bondy and Murty [2], Harary [9], Tutte [22], and Xu [23] have done much work in the field of graph theory. In classic graph theory, the edges and the vertices are all deterministic. However, in practical application, as the system becomes more complex, indeterminacy factors may appear in graphs. If the weight of each edge is of indeterminacy, it is not suitable to employ the traditional methods to study the weighted matching problem. Sometimes, whether two vertices are joined by an edge cannot be completely determined in a graph. As a result, the traditional methods can not be used to verify some properties of the graph. In order to deal with indeterminacy factors in graph, random graphs were first defined by Erd¨ os and R´ enyi [4, 5]. They thought that whether two vertices are joined can be described as a random variable. Since then, the random graph has been studied by many researches, such as Bollob´ as [1], Gilbert [8], Mahmoud et al. [19], etc. Unfortunately, it is not suitable to employ the probability theory to deal with every indeterminacy phenomenon. A fundamental premise of applying probability theory is that the sample size is large enough, and the estimated probability is close enough to the real frequency. However, in the real life, we are frequently lack of observed data about the unknown state of nature, not only for technical difficulties, ∗ Corresponding

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but also for economic reasons. Then, how does one deal with this kind of indeterminacy factors? Usually, we have to invite some domain experts to evaluate the belief degree that each event will occur. Since human beings usually overweight unlikely events (Kahneman and Tversky [10]), the belief degree has much larger variance than the real frequency. In this case, the probability theory is no longer valid. In order to deal with this indeterminacy phenomenon distinguishing from randomness, Liu [12] founded the uncertainty theory in 2007. Up to now, the uncertainty theory has become a branch of mathematics for modeling human uncertainty. In order to describe dynamic uncertain systems, Liu [13] introduced uncertain process. Besides, uncertain calculus was initialized by Liu [14] to deal with differentiation and integration of functions of uncertain processes. Furthermore, You [24] proved some convergence theorems of uncertain sequences, Liu and Xu [18] studied some properties on uncertain variables, and Peng and Iwamura [21] given a sufficient and necessary condition of uncertainty distribution. In practical aspect, the uncertainty theory was first introduced into finance by Liu [14]. In addition, Peng and Yao [20] proposed an uncertain stock model and given the corresponding option pricing formulas. Furthermore, Gao [7] employed the uncertainty theory to investigate the shortest path problem with uncertain arc lengths. Zhang and Peng [25] presented some uncertain programming models for Chinese postman problem in uncertain environment. For exploring the recent developments of the uncertainty theory, the readers may consult Liu [17]. Gao and Gao [6] first defined the uncertain graph. As a powerful tool for modeling human uncertainty, the uncertainty theory is employed to deal with uncertain factors in graph. Recently, Zhang and Peng [26] given an Euler index to show how likely an uncertain graph is Eulerian. An uncertain graph refers to the graph in which whether two vertices of the graph are joined by an edge cannot be completely determined. Then how to verify the properties of the graph? Matching is one of the basic concept of the graph theory. For a given uncertain graph, at how much belief degree we can regard the uncertain graph has a maximum matching? In this paper, the concepts of matching index and perfect matching index of uncertain graph are firstly proposed. With the framework of uncertainty theory, an algorithm to calculate matching index of uncertain graph is given. The remainder of this paper is organized as follows. After introducing some basic concepts and properties of uncertainty theory and graph theory in Section 2, Section 3 proposes the concepts of matching index and perfect matching index of uncertain graph, and also investigates some properties of the matching index. In Section 4, an algorithm to calculate the matching index of uncertain graph is given. The last section concludes this paper with a brief summary.

2

Preliminaries

2.1

Uncertainty Theory

Now, we present some preliminaries from uncertainty theory. Let Γ be a nonempty set, and L a σ-algebra over Γ. Each element Λ ∈ L is called an event. The set function M{Λ} is called an uncertain measure if it satisfies the following three axioms (Liu [12]):

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(1)(Normality Axiom) M{Γ} = 1; (2)(Duality Axiom) M{Λ} + M{Λc } = 1 for any Λ ∈ L; (3)(Subadditivity Axiom) For every countable sequence of events {Λi }, we have (∞ ) ∞ [ X M Λi ≤ M{Λi }. i=1

i=1

Uncertain measure is one of the basic concept of the uncertainty theory that is used to indicate the belief degree that an uncertain event may occur. The triplet (Γ, L, M) is called an uncertainty space. In order to obtain an uncertain measure of compound event, the fourth axiom called product axiom was presented by Liu [14]. (4)(Product Axiom) Let (Γk , Lk , Mk ) be uncertainty spaces for k = 1, 2, · · · The product uncertain measure M is an uncertain measure satisfying (∞ ) ∞ Y ^ M Λk = Mk {Λk } k=1

k=1

where Λk are arbitrarily chosen events from Lk for k = 1, 2, · · · , respectively. An uncertain variable is a measurable function ξ from an uncertainty space (Γ, L, M) to the set of real numbers. Liu [12] introduced the independence concept of uncertain variables. The uncertain variables ξ1 , ξ2 , · · · , ξn are said to be independent if ( n ) n ^ \ M{ξi ∈ Bi } M (ξi ∈ Bi ) = i=1

i=1

for any Borel sets B1 , B2 , · · · , Bn of