Euler Index in Uncertain Graph

Euler Index in Uncertain Graph Bo Zhang 1, Jin Peng 2, 1 School of Mathematics and Statistics, Huazhong Normal University Hubei 430079, China 2 Institute of Uncertain Systems, Huanggang Normal University Hubei 438000, China Abstract As the complexity of a system increases, in practical application of graph theory, different types of uncertainty are frequently encountered In an uncertain graph, whether two vertices of the graph are joined cannot be completely determined Within the framework of uncertainty theory, the concept of Euler index of uncertain graph is proposed It also gives a method to calculate Euler index of uncertain graph What s more, the Euler index of uncertain cycle and uncertain graph with blocks can be obtained in a simple way Keywords: uncertain graph, uncertainty theory, Euler index 1 Introduction It is quite well known that graphs are simply models of relations A graph is a convenient way of representing information involving relationship between objects The objects are represented by vertices and relations by edges For example, the vertices could represent people, with edges joining pairs of friends; or the vertices might be communication centers, with edges representing communication links In mathematics and computer science, graph theory is the study of graphs The paper written by Euler on the seven bridges problem and published in 1736 is regarded as the first paper in the history of graph theory More than one century after Euler s paper on the seven bridges problem, the concept of a tree, a connected graph without cycles, appeared implicitly in the work of Kirchhoff Later, Cayley, Polya, and others used tree to enumerate chemical molecules One of the most famous and productive problems of graph theory is the four color problem It was first posed by Guthrie in 1852, and many celebrated incorrect proofs have been proposed, including those by Cayley, Kempe, and others The four color problem remained unsolved for more than a century In 1976, it was proved by Appel and Haken [1][2] Some researchers, such as Dirac [12], Harary [18], Woodall [36], Edmonds and Johnson [13], Bondy and Murty [10], Bermond and Thomassen [4], Xu [37], have done much work in the field of graph theory In classical graph theory, the edges and the vertices are all deterministic However, as the complexity of a system increases, different types of uncertainty are frequently encountered in practical application As a result, many uncertain factors appear in graphs, which leads to some uncertain situations Sometimes, whether two vertices are joined by an edge cannot be completely determined Then, how to deal with these uncertain factors? Some researchers introduced probability theory into the graph theory Random graphs were first defined by Erdös and Rényi [14] At nearly the same time, Gilbert [17] studied the probability that the random graph is connected, and also the probability that two specific vertices are connected Later, Corresponding author E-mail addresses: pengjin01@tsinghuaorgcn 1

Bollobás [8], investigated the degree sequences of random graphs, in which the edges are chosen independently and with the same probability Furthermore, Luczak [27] studied the behavior of a random graph process Many other researchers, such as Mahmoud et al [28], Barabási and Albert [3], Bollobás et al [9], have done a lot of work in this field In 1965, Zadeh [38] introduced the concept of fuzzy sets in his classical paper After that, Rosenfeld [32] introduced fuzzy graphs in 1975 Since then, lots of works on fuzzy graphs have been carried out For instance, Bhattacharya [5] considered some properties of fuzzy graphs, and introduced the notions of eccentricity and center In 1994, Mordeson and Peng [30] defined the operations of Cartesian product, composition, union, and join on fuzzy subgraphs of graphs G 1 and G 2 In 2003, Bhutani and Battou [6] considered operations on fuzzy graphs under which M-strong property is preserved For more research of random graphs, we may consult Bhutani and Rosenfeld [7], Sameena and Sunitha [33], Mathew and Sunitha [29], etc However, if uncertain factor comes from the decision-maker s empirical estimation, it is not suitable to employ random variable or fuzzy variable to describe this kind of uncertain factor In 2007, Liu [19] proposed uncertainty theory, which has become a branch of axiomatic mathematics Liu [25] wrote that when the sample size is too small (even no-sample) to estimate a probability distribution, we have to invite some domain experts to evaluate their belief degree that each event will occur Since human beings usually overweight unlikely events, the belief degree may have much larger variance than the real frequency and then probability theory is no longer valid In this situation, we should deal with it by uncertainty theory It is too adventurous if we deal with the belief degree by probability theory, because it may lead to counterintuitive results From then on, uncertainty theory provides a powerful tool to deal with uncertain in graph In 2011, Gao and Gao [16] proposed the concept of uncertain graph, and investigate the connectedness index of uncertain graph In an uncertain graph, whether two vertices of the graph are joined cannot be completely determined Then, to an uncertain graph, in how much belief degree we can regard the graph is Eulerian? In this paper, under the framework of uncertainty theory, the concept of Euler index in uncertain graph is proposed Also, a method to calculate Euler index is given In addition, we find that the Euler index of uncertain cycle and uncertain graph with blocks can be obtained in a simple way The remainder of this paper is organized as follows In Section 2, some basic concepts and properties of uncertainty theory and uncertain graph used throughout this paper are introduced In Section 3, the concept of Euler index in uncertain graph is proposed After that, a method to calculate the Euler index is given Furthermore, the Euler index of uncertain cycle and uncertain graph with blocks are investigated in Section 4 and Section 5, respectively Finally, we conclude the paper in Section 6 2 Preliminaries 21 Uncertainty Theory Founded by Liu [19] in 2007 and refined by Liu [24] in 2010, uncertainty theory has become a branch of mathematics for modeling human uncertainty Liu [22] presented uncertain programming which is a type of mathematical programming involving uncertain variables, and applied uncertain programming to industrial engineering and management science In addition, uncertain process was defined by Liu [20] as a sequence of uncertain variables indexed by time or space Furthermore, uncertain calculus was initialized by Liu [21] to deal with differentiation and integration of functions of uncertain processes Based on uncertain calculus, Liu [20] proposed a tool of uncertain differential equations After that, Chen and Liu [11] proved the existence and uniqueness theorem for uncertain differential equations In addition, uncertainty theory was also applied to uncertain statistics (Liu [24], Wang et al [34][35]), uncertain inference (Liu [23], Gao et al [15]), uncertain control (Liu [23], Zhu [39]), and uncertain finance (Liu [21], Peng and Yao [31]) 2

In this section, we present some basic concepts and results from uncertainty theory, which will be used throughout this paper Let Γ be a nonempty set, and L a σ-algebra over Γ Each element Λ L is called an event For any Λ L, M{Λ} is a function from L to [0, 1] In order to ensure that the number M{Λ} has certain mathematical properties, Liu [19][24] presented the following four axioms: normality, duality, subadditivity, and product axioms If the first three axioms are satisfied, the function M{Λ} is called an uncertain measure The triplet (Γ, L, M) is called an uncertainty space Definition 1 [19] An uncertain variable is a measurable function ξ from an uncertainty space (Γ, L, M) to the set of real numbers, ie, for any Borel set B of real numbers, the set is an event {ξ B} = {γ Γ ξ(γ) B} The uncertainty distribution of an uncertain variable ξ is defined by Φ(x) = M{ξ x} for any real number x For example, the zigzag uncertain variable ξ Z(a, b, c) has an uncertainty distribution 0, if x a x a 2(b a), if a x b Φ(x) = x + c 2b 2(c b), if b x c 1, if x c Definition 2 [19] Let ξ be an uncertain variable Then the expected value of ξ is defined by E[ξ] = + M{ξ r}dr 0 0 provided that at least one of the two integrals is finite M{ξ r}dr If ξ has an uncertainty distribution Φ, then the expected value may be calculated by E[ξ] = + (1 Φ(x))dx 0 0 Φ(x)dx Theorem 1 [24] Let ξ 1, ξ 2,, ξ n be independent uncertain variables with uncertainty distributions Φ 1, Φ 2,, Φ n, respectively If the function f(x 1, x 2,, x n ) is strictly increasing with respect to x 1, x 2,, x m and strictly decreasing with respect to x m+1, x m+2,, x n, then ξ = f(ξ 1, ξ 2,, ξ n ) is an uncertain variable with inverse uncertainty distribution Ψ 1 (α) = f(φ 1 1 (α),, Φ 1 m (α), Φ 1 m+1 (1 α),, Φ 1 n (1 α)) Furthermore, Liu and Ha [26] proved that the uncertain variable ξ = f(ξ 1, ξ 2,, ξ n ) has an expected value E[ξ] = 1 0 f(φ 1 1 (α),, Φ 1 m (α), Φ 1 m+1 (1 α),, Φ 1 n (1 α))dα A function is said to be Boolean if it is a mapping from {0, 1} n to {0, 1} An uncertain variable is said to be Boolean if it takes values either 0 or 1 3

Theorem 2 [24] Assume that ξ 1, ξ 2,, ξ n are independent Boolean uncertain variables, ie, { 1, with uncertain measure α i ξ i = 0, with uncertain measure 1 α i for i = 1, 2,, n If f is a Boolean function, then ξ = f(ξ 1, ξ 2,, ξ n ) is a Boolean uncertain variable such that sup min ν i(x i ), if sup min ν i(x i ) < 05 f(x M{ξ = 1} = 1,x 2,,x n)=11 i n f(x 1,x 2,,x n)=11 i n 1 sup min ν i(x i ), if sup min ν i(x i ) 05 1 i n 1 i n f(x 1,x 2,,x n)=0 where x i take values either 0 or 1, and ν i are defined by { α i, if x i = 1 ν i (x i ) = 1 α i, if x i = 0 for i = 1, 2,, n, respectively f(x 1,x 2,,x n)=1 22 Uncertain Graph In classic graph theory, the edges and vertices are all deterministic, either exist or not The number of vertices in G is often called the order of G, while the number of edges is called its size Assume G is a graph of order n, then the adjacency matrix of G is the n n matrix d 11 d 12 d 1n d 21 d 22 d 2n D = d n1 d n2 d nn where d ij = { 1, if there exists an edge between vertices i and j 0, otherwise However, in application, some uncertain factors will appear because of the lack of observed data, insufficient information or some other reasons Usually, we obtain the belief degree that each event will occur by means of expert s empirical estimation Could we deal with the belief degree by probability theory when we are lack of observed data? Assume that the vertices represent people, with edges joining pairs of friends in graph, and whether two people are friends is not exactly known Let us imagine what will happen if we regard the event that whether two people are friends as a random event When two people are friends with probability 05, and not friends with probability 05 if we regard the event that whether two people are friends as a random event Assume there are 100 people, since there do not exist any observed information for two people are friends, we have to regard the probability of any two people are friends as iid random variables Therefore, we should have Pr{any two people are not friends} 0, Pr{any two people are friends with each other} 0 However, whether two people are friends is in fact invariant Thus one and only one of the following alternatives holds: (a) any two people are not friends, (b) any two people are friends with each other 4

This result dose not coincide with the theoretical probability analysis that says both consequences are almost impossible Hence we cannot regard the event that whether two people are friends as a random event because it is invariant, only the real value cannot be exactly observed However, if two people are friends with some belief degrees in uncertain measure, then we have M{any two peoples are not friends} = 05, M{any two people are friends with each other} = 05 Do you think the uncertainty theory provides a reasonable explanation about the result of experiment? Obviously, in real life situations, uncertain factors will no doubt appear in graphs When there is imprecision in the description of the relationships of the objects, it is natural that we need to define uncertain graph In 2011, Gao and Gao [16] proposed the concept of uncertain graph in which all edges are independent and exist with some belief degrees in uncertain measure In other words, the number α ij in the uncertain adjacency matrix indicate the edge between vertices i and j exist with uncertain measure α ij and dose not exist with uncertain measure 1 α ij Definition 3 [16] A graph of order n is said to be uncertain if it has uncertain adjacency matrix α 11 α 12 α 1n α 21 α 22 α 2n α n1 α n2 α nn where α ij represent that the edges between vertices i and j exist with uncertain measures α ij, i, j = 1, 2,, n, respectively Generally speaking, we assume α ii = 0 for i = 1, 2, n because there is no edge between any vertex and itself Note that, if the uncertain graph is undirected, then the uncertain adjacency matrix is symmetric, ie, α ij = α ji for any i and j Notice that in such graphs one is mainly interested in whether or not two given vertices are joined by an edge; the manner in which they are joined is immaterial In order to show how likely an uncertain graph is connected, a connectedness index is defined as follows Definition 4 [16] The connectedness index of an uncertain graph is the uncertain measure that the uncertain graph is connected A key problem for us is how to obtain the connectedness index when an uncertain graph is given Then, the following theorem was proposed to solve this problem Theorem 3 [16] Assume an uncertain graph G has an uncertain adjacency matrix α 11 α 12 α 1n α 21 α 22 α 2n α n1 α n2 α nn If all edges are independent, then the connectedness index is sup min ν ij(x), if sup f(x)>1 ρ(g) = 1 sup min ν ij(x), if sup f(x) 0 f(x)>1 f(x)>1 min ν ij(x) < 05 min ν ij(x) 05 5

where x 11 x 12 x 1n x 21 x 22 x 2n X =, x n1 x n2 x nn and x ij take values either 0 or 1, and ν ij are defined by { α ij, if x ij = 1 ν ij (X) = 1 α ij, if x ij = 0 for i, j = 1, 2,, n, respectively, and f(x) = I + X + X 2 + + X n 1 3 Euler Index In graph theory, a tour of G is a closed walk that traverses each edge of G at least once An Euler tour is a tour which traverses each edge exactly once A graph is Eulerian if it contains an Euler tour, and non-eulerian otherwise Also, there exists a necessary and sufficient condition to determine whether a graph is Eulerian: A nonempty connected graph is Eulerian if and only if it has no vertices of odd degree The degree of a vertex v in G is the number of edges of G incident with v In order to show how likely an uncertain graph is Eulerian, an Euler index is defined below Definition 5 The Euler index of an uncertain graph is the uncertain measure that the uncertain graph is Eulerian How can we calculate the Euler index when an uncertain graph is given? completely solves this problem The following theorem Theorem 4 Let G be an uncertain graph of order n and its uncertain adjacency matrix is α 11 α 12 α 1n α 21 α 22 α 2n α n1 α n2 α nn If all edges are independent, then the Euler index of G is sup min ν ij(x ij ), if sup g(x)=1 µ(g) = 1 sup min ν ij(x ij ), if sup g(x)=0 g(x)=1 g(x)=1 min ν ij(x ij ) < 05 min ν ij(x ij ) 05 where X = x 11 x 12 x 1n x 21 x 22 x 2n, x n1 x n2 x nn 6

x ij {0, 1}, and ν ij are defined by ν ij (x ij ) = { α ij, if x ij = 1 1 α ij, if x ij = 0 for i, j = 1, 2,, n, respectively, and I + X + X 2 + + X n 1 > 0 1, if g(x) = x ij are even for i = 1, 2,, n 1 j n 0, otherwise Proof by Note that all edges are essentially independent Boolean uncertain variables, and are represented for i, j = 1, 2,, n Write ξ ij = { 1, with uncertain measure α ij 0, with uncertain measure 1 α ij ξ 11 ξ 12 ξ 1n ξ 21 ξ 22 ξ 2n Ξ = ξ n1 ξ n2 ξ nn Then the uncertain graph is connected if and only if I + Ξ + Ξ 2 + + Ξ n 1 > 0 And we know that the degree of a vertex i is just the sum of entries in the row corresponding to vertex i in Ξ Then the uncertain graph is Eulerian if and only if Thus according to Definition 5, the Euler index is g(ξ) = 1 µ(g) = M{g(Ξ) = 1} Since the function g is Boolean, it follows from Theorem 2 that the theorem is proved Assume G is an uncertain graph of order n and size m, Theorem 4 provides a method to obtain Euler index of G Roughly speaking, the method can be summarized as follows: Step 1: Set S = {φ}, k = 0, µ = 0, µ = 0 If there exist an adjacency matrix X k such that G is Eulerian Go to Step 2 Otherwise, stop, µ(g) = 0 Step 2: Calculate min ν ij(x k ) Set µ = sup{µ, min ν ij(x k )}, S = S {X k } and k = k + 1 Step 3: If there exist other adjacency matrix X k S such that G is Eulerian, go to Step 2 otherwise, go to Step 4 Step 4: If µ < 05, stop, and µ(g) = µ Otherwise, go to Step 5 Step 5: Choose an arbitrary adjacency matrix X k S, calculate min ν ij(x k ) Set µ = sup {µ, min ν ij(x k )}, S = S {X k } and k = k + 1 Step 6: If k = 2 m + 1 stop, and µ(g) = 1 µ Otherwise, go to Step 5 Example 1: Let G be an uncertain graph of order 3 and size 3 has an uncertain adjacency matrix 0 08 06 08 0 07 06 07 0 7

Since the size of uncertain graph G is 3, its adjacency matrix breaks down into eight cases Assume adjacency matrix is 0 1 1 1 0 1 1 1 0 Then the graph G is Eulerian, ie, g(x) = 1, and sup g(x)=1 min ν ij(x) = 06 1 i,j 3 Assume adjacency matrix is one of the following seven matrices 0 0 0 0 1 0 0 0 0 0 0 0, 1 0 0, 0 0 1, 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 0, 0 0 1, 0 0 0, 1 0 1 1 0 0 1 1 0 1 0 0 0 1 0 Then the graph G is non-eulerian, ie, g(x) = 0, and sup g(x)=0 It follows from Theorem 4 that the Euler index is µ(g) = 1 min ν ij(x) = 04 1 i,j 3 sup g(x)=0 min ν ij(x) = 06 1 i,j 3 Example 2: Let G be an uncertain graph of order 6 and size 15 has an uncertain adjacency matrix 0 08 05 08 09 02 08 0 07 08 04 03 05 07 0 06 04 06 08 08 06 0 07 04 09 04 04 07 0 03 02 03 06 04 03 0 Employ the method mentioned above, we get the Euler index µ(g) = 04, which is obtained by MATLAB In addition, we have the following Corollary 1 Corollary 1 Let G be an uncertain graph, then the Euler index of G is less than or equal to the connectedness index of G, ie, µ(g) ρ(g) Example 3: Assume an uncertain graph G has an uncertain adjacency matrix presented in Example 1 We have known that the Euler index of G is µ(g) = 06 It follows from Theorem 3 that the connectedness index of G is ρ(g) = 07 8

Example 4: Assume an uncertain graph G has an uncertain adjacency matrix presented in Example 2 We have known that the Euler index of G is µ(g) = 04 It follows from Theorem 3 that the connectedness index of G is ρ(g) = 06 Example 5: Assume an uncertain graph G has an uncertain adjacency matrix 0 08 06 08 0 06 06 06 0 It follows from Theorem 4 and Theorem 3 that the Euler index of G is equal to the connectedness index of G, ie, µ(g) = ρ(g) = 06 4 Euler Index of Uncertain Cycle A walk is closed if it has positive length and its origin and terminus are the same A cycle is a closed trail whose origin and internal vertices are distinct In this paper, we use the term uncertain cycle to denote an uncertain graph corresponding to a cycle Note that any uncertain cycle s uncertain adjacency matrix can be denoted as below 0 α 12 0 0 α 1n α 12 0 α 23 0 0 0 α 32 0 α 34 0 0 0 0 α n 2n 3 0 α n 2n 1 0 0 0 α n 1n 2 0 α n 1n α n1 0 0 α nn 1 0 The following theorem tells us how to obtain the Euler index of uncertain cycle in a simple way Theorem 5 Let G be an uncertain circle of order n, denote A as the uncertain adjacency matrix Then the Euler index of G is the smallest positive value of α ij, for i, j = 1, 2,, n, respectively Proof Assume α is the smallest positive value of α ij, for i, j = 1, 2,, n, respectively Then we have that sup min ν ij(x ij ) = α g(x)=1 According to Theorem 4, we consider the following two cases Case 1 If α < 05 µ(g) = α Case 2 If α 05 sup min ν ij(x ij ) = 1 α Thus g(x)=0 Hence the theorem is proved µ(g) = 1 sup g(x)=0 min ν ij(x ij ) = α Example 6: Assume an uncertain cycle G has an uncertain adjacency matrix presented in Example 1 It follows from Theorem 5 that the Euler index of G is µ(g) = 06 9

Example 7: Assume an uncertain cycle G has an uncertain adjacency matrix 0 03 0 0 0 05 03 0 06 0 0 0 0 06 0 04 0 0 0 0 04 0 06 0 0 0 0 06 0 07 05 0 0 0 07 0 It follows from Theorem 5 that the Euler index of G is µ(g) = 03 5 Euler Index of Uncertain Graph with Blocks Generally, a graph can be denoted by G(V, E), where V is a set of vertices, E is a set of edges Suppose that V is a nonempty subset of V The subgraph of G obtained from G by deleting the vertices in V together with their incident edges, and is denoted by G V If V = {v} we write G v for G {v} Assume G is connected, a cut vertex of G is a vertex of V such that G {v} is disconnected A connected graph that has no cut vertex is called a block A block of a graph is a subgraph that is a block and is maximal with respect to this property, this is illustrated in Figure 1 v 4 v 1 v 2 v 3 v 7 v 5 v 6 G 1 G 2 Figure 1: Graph G with blocks G 1 and G 2 There exists a conclusion in graph theory that if G is Eulerian, then every block of G is Eulerian Then we have the following Theorem 6 Theorem 6 Let G be an uncertain graph with blocks G 1, G 2,, G k, then the Euler index of G is the smallest Euler index of G i, i = 1, 2,, k, respectively, ie, µ(g) = µ(g 1 ) µ(g 2 ) µ(g k ) Proof The theorem follows from the fact that if G is Eulerian, then every block of G is Eulerian Theorem 6 provides a new method to obtain Euler index of uncertain graph with blocks G 1, G 2,, G k Generally speaking, the method can be summarized as follows: Step 1: Calculate the Euler index of G i, ie, µ(g i ), i = 1, 2,, k, respectively 10

Step 2: The Euler index of G is µ(g) = µ(g 1 ) µ(g 2 ) µ(g k ) Example 8: Let G be an uncertain graph as shown in Figure 1, and it has an uncertain adjacency matrix 0 08 02 07 0 0 0 08 0 07 04 0 0 0 02 07 0 09 07 04 08 07 04 09 0 0 0 0 0 0 07 0 0 09 02 0 0 04 0 09 0 07 0 0 08 0 02 07 0 It follows from Theorem 4 that the Euler index of G 1 and G 2 are µ(g 1 ) = 06, µ(g 2 ) = 07 Thus, the Euler index of G is µ(g) = µ(g 1 ) µ(g 2 ) = 06 Example 9: Let G be an uncertain graph as shown in Figure 1, and it has an uncertain adjacency matrix 0 08 02 07 0 0 0 08 0 07 04 0 0 0 02 07 0 09 02 04 03 07 04 09 0 0 0 0 0 0 02 0 0 01 03 0 0 04 0 01 0 02 0 0 03 0 03 02 0 It follows from Theorem 4 that the Euler index of G 1 and G 2 are µ(g 1 ) = 06, µ(g 2 ) = 02 Thus, the Euler index of G is µ(g) = µ(g 1 ) µ(g 2 ) = 02 6 Conclusion In the applications of graph theory, uncertain factors will no doubt appear in graphs This paper concerns about Euler tour in uncertain graph in which all edges are independent and exist with some belief degrees in uncertain measure To show how likely an uncertain graph is Eulerian, an Euler index is proposed, and then the method to calculate the Euler index is proposed Besides, the Euler index of uncertain cycle and uncertain graph with blocks can be obtained in a simple way Acknowledgements This work is supported by the National Natural Science Foundation (No60874067), the Hubei Provincial Natural Science Foundation (No2010CDB02801), and the Scientific and Technological Innovation Team Project (NoT201110) of Hubei Provincial Department of Education, China 11

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