Abstract
There has been an upsurge of research on complex networks in recent years. The purpose of this paper is to study the mathematical properties of the random pentagonal chain networks with the help of graph theory. Based on the networks , we first obtain the expected value expressions of the Gutman index, Schultz index, multiplicative degree-Kirchhoff index, and additive degree-Kirchhoff index, and then, we get the explicit expression formulas of their variances. Finally, we find that their limiting distributions all have the probabilistic and statistical significance of normal distribution.
1. Introduction
Complex networks have become increasingly important in scientific research, especially in statistical physics and information science [1]. In the research of complex networks, the analysis and synthesis issues of complex dynamic networks have received extensive attention in recent years [2]. Therefore, researchers began to try to use this new theoretical tool to study some complex systems in real life, among which the structure of complex systems and the relationship between system structure and function are the focuses of researches [3, 4]. The application of graph theory, as the name implies, is to solve the corresponding problems in real life by combining the knowledge of graph theory, widely used in engineering design. Graph theory is a basic but very effective tool to study complex networks. There are a lot of complex systems in nature which can be described by network graphs. It is composed of many nodes and the edges between the nodes, where the nodes represent different individuals and the edges represent the relationships between individuals. Topological index is one of the tools in graph theory, which can reflect the related properties and connected information of some kinds of topological structures. It has been extensively researched in graph theory. Therefore, this paper combines the three concepts to make a quantitative analysis of a kind of random pentagonal chain networks.
Suppose that represents a simple undirected graph with edge set and vertex set . In , the number of edges associated with a vertex is called the degree of it, and we express it by . The distance between any two vertices and in the graph represents the length of the shortest path between them. To learn more about the relevant definitions, one can be referred to [5]. The Wiener index is the sum of the distances between all pairs of vertices in a graph , denoted by
This graph invariant related to the distance of a graph was introduced into chemistry in 1947 [6] and mathematics 30 years later [7]. It is a very important parameter in certain applications of chemistry and network theory, demonstrating the connectivity of the structure. Binu et al. [8] discussed the Wiener index of a fuzzy graph and application to illegal immigration networks. Asir and Rabikka [9] presented a method for calculating the Wiener index of zero-divisor graph of any positive integer . Wei and Shiu [10] obtained the expected values and the average values of the Wiener index in random polygonal chains. Chen et al. [11, 12] studied and compared the Wiener index in the catacondensed hexagonal system and the tree-like polyphenyl system. At present, the Wiener index is still broadly used in the fields of physics, molecular biology, and mathematics. For more applications, please refer to [13–18].
Based on the concept of distance and electrical network theory, the resistance distance is derived, which was put forward by Klein and Randić [19]. When each edge in the graph is equivalent to a unit resistance, it can be considered as the effective resistance between two vertices. We can learn more details through [20]. The Kirchhoff index , also known as full effective resistance, or effective graph resistance, is defined as the sum of the resistance distances of all vertex pairs in the graph [21–23], that is,
Through the efforts of the researchers, it has been founded that the Kirchhoff index has played an important role in chemistry. For linear hexagonal chains, some special molecular graphs, such as the bipartite unicyclic graphs, bicyclic graphs, and Cayley graphs, we can calculate the Kirchhoff indices of these polycyclic structures to evaluate their cyclicity [24–28]. At the same time, the Kirchhoff index is also of great significance in mathematics. For the details of the mathematical properties, please refer to [29–31].
As for the Wiener index, researchers improved it by weighting it. That is, a graph related to the weight function , expressed by . Suppose represents a calculation symbol in , , , and , then the transformed Wiener index can be expressed as
When is the arithmetic operation and set , the Gutman index is obtained. Then, Equation (3) is equivalent to
When is the arithmetic operation and set , the Schultz index is obtained. Then, Equation (3) is equivalent to
In the field of chemistry, the Gutman index and Schultz index can accurately reflect the similar molecular structure characteristics as the Wiener index. For more details, properties, and applications of the Gutman index and Schultz index, if you are interested, you can refer to the paper [32–37].
Similarly, researchers also improved and optimized the Kirchhoff index. Considering the degree of vertices in a graph, the multiplicative degree-Kirchhoff index was put forward in 2007 by Chen and Zhang [38], which was expressed as
In 2012, the additive degree-Kirchhoff index was proposed by Gutman et al. [39], denoted by
The Kirchhoff index and degree-Kirchhoff index are used as graph parameters based on resistance distance in physics and chemistry. And network science also has a wide range of applications. For an electrical network, the smaller the Kirchhoff index is, the less power the network consumes per unit time. Chemically, Kirchhoff index can be used to describe the structural characteristics of molecules and define the topological radius of polymers. In network science, the smaller the Kirchhoff index is, the stronger the robustness of the network is. These fields have attracted the attention of more and more researchers. Readers can refer to references [40–42] for more information.
The random chain networks studied in this paper are finite 2-connected network graphs, which can be seen as being formed by sequentially connecting pentagons. That is, is obtained by and then randomly connected a pentagon at the end, as shown in Figure 1. When, as shown in Figure 2. For , there are two ways to add pentagons at the end, and the results can be expressed as and , as shown in Figure 3. For such a random chain networks, any step for is stochastic, and their probabilities are and , respectively, that is, from to with probability and to with probability , in which the probabilities and are constants and independent to the step at the same time.
Motivated by [43], we use two random variables and to represent our choices. If our choice is , we put ; otherwise, it is equal to 0 (). One holds that and .
Yang and Zhang [44] and Ma et al. [45] determined explicit formulas of and for a class of random chain networks composed of hexagons, respectively. Similarly, Huang et al. [46] obtained for a class of spiro random chain networks composed of hexagons. Wei and Shiu [10] put forward the expression of in random polygonal chains and proved its asymptotic property. At the same way, Zhang et al. [47] obtained the simple formulas of , , , and for a class of random polyphenylene chain networks.
The research motivation of this paper is to investigate the structural properties of random pentagonal chain networks and compare the similarities and differences between their corresponding different topological indices. Motivated by [43, 45–48], for this paper, in Section 2, we determine the explicit formulas of , , , and for the random pentagonal chain networks and show their differences visually through the function images. In Section 3, we continue to calculate , , , and and to prove that the four indices of asymptotically obey normal distributions. In order to accomplish the purpose of this paper, we must put forward the following hypotheses.
Hypothesis 1. It randomly and independently chooses a way attaching the new terminal pentagonto,. To be more precise, the sequences of random variables are independent and must satisfy Equation (8).
Hypothesis 2. We put .
Under the condition that both Hypotheses 1 and 2 are true, we can get the following:
(1)The analytical expressions of the variances of , , , and are obtained(2)When , we verify that the random variables , , , and asymptotically obey normal distributions. Replace them with . It is evident to see thatwhere and represent the expectation and variance of this random variable , respectively
In this paper, assume that and are two functions of . We put if and put if .
2. The Expected Values of , , , and
For the random chain networks , we find that , , , and are random variables. Then, we determine the analytic expressions of , , , and in this section.
In fact, is organized by adding a new terminal pentagon to by an edge, where the vertices of are labelled as , , , , and in clockwise direction. For all , one has
Meanwhile,
Theorem 3. For , the analytic expression of is
Proof of Theorem 3. By Equation (4), we use to denote the set of all other vertices in after the vertex is removed. Then, one can be convinced that
Note that
Recall that and for . From Equation (10), we have
From Equation (11), one follows that
Then,
For the random chain networks , we obtain that is a stochastic variate. Let
By using the above formula and Equation (17), we can get the following relation for . One sees
Then, we go on to consider the following two possible cases.
Case 1. .
In this case, (of ) overlaps with or (of ). Therefore, is rewritten as or with probability .
Case 2. .
In this case, (of ) overlaps with or (of ). Therefore, is rewritten as or with probability .
Together with the above cases, by applying the expectation operator and Equation (10), one follows that where
Then, we obtain
Meanwhile, for , the boundary condition is
Using above condition and the recurrence relation with respect to , it is no hard to obtain
From Equation (19), it holds that
For , we obtain . Similarly, according to the recurrence relation related to , we have
Theorem 4. For , the analytic expression of is
Proof of Theorem 4. Notice that the random chain networks are organized by adding a new terminal pentagon to by an edge. By Equation (5), one has where Let . It is routine to check that
Because has vertices and , for . By Equation (10), we can know
Since that, for . From Equation (11), one sees that
Then, Equation (28) can be rewritten as
We know that is a random variable. Let
According to the above formula and Equation (33), we have the following relation for . It holds that
We proceed by taking into account the following two cases.
Case 1. .
In this case, (of ) coincides with or (of ). Then, is given by or with probability .
Case 2. .
In this case, (of ) coincides with or (of ). Then, is given by or with probability .
Together with Case 1 and Case 2, we obtain where
Then, as an immediate consequence, we have
When , we can easily find that
Then, using above formula and the recurrence relation, it is routine to check that
By using Equation (24) and Equation (35), we can get
According to the initial conditions of the difference equation, we find . Then, we arrive at
Next to start, we determine the analytic expressions of and . We continue to follow the original conditions, is organized by adding a new terminal pentagon to by an edge, where the vertices of are labelled as , , , , and in clockwise direction. For all , one has
Therefore,
Theorem 5. For , the analytic expression of is
Proof of Theorem 5. By Equation (6), one can be convinced that Note that Because of and for , from Equation (43), we have From Equation (44), we follow that Then, For the , we know that is a stochastic variate. Let
Then, let us continue to consider the two probable cases shown below.
Case 1. .
In this case, (of ) overlaps with or (of ). Therefore, is rewritten as or with probability .
Case 2. .
In this case, (of ) overlaps with or (of ). Therefore, is rewritten as or with probability .
Together with the above cases, by applying the expectation operator and Equation (43) and Equation (44), we can get that
By taking the expectation of both sides of the above equation, and noting that , we obtain
When , the boundary condition is
By using the above results to solve the difference equation, we can obtain
So combining formulas (50) and (55), we can know
We get that for . Similarly, can be obtained by solving the equation
Theorem 6. For , the analytic expression of is
Proof of Theorem 6. According to the known conditions and the composition rules of the networks , by Equation (7), one has where Let . It is routine to check that Note that has vertices and and for . By Equation (43), we can get And due to for , from Equation (44), one sees that Then, Equation (59) can be rewritten as
For that reason,is a random variable. Let
And we proceed by taking into account the following two cases.
Case 1. .
In this case, (of ) coincides with or (of ). Then, is given by or with probability .
Case 2. .
In this case, (of ) coincides with or (of ). Then, is given by or with probability .
Together with Case 1 and Case 2, we have
By taking the expectation of both sides of the above equation, and noting that , we obtain
When is equal to 1, the initial condition is
Combining the above conditions, through the calculation, we can get
From Equations (64) and (69), we can get
Then according to and the recurrence relation related to , we have
Next, we use the computer to generate the images of expectation functions of the above four topological indices about the networks , as shown in Figure 4.
Remark 7. Through the figure, we can see that the expected values of the four indices are related to two variables and , and all of them show a positive correlation. Among them, the growth rate of is the fastest and grows the slowest, while the growth rates of and are between them. This further illustrates the relationships between these four topological indices and vertex distance and degree.
3. The Variances and Limiting Behaviours for , , , and
In this section, we will calculate the explicit analytical expressions for , , , and . And we will prove that the Gutman, Schultz, multiplicative degree-Kirchhoff, and additive degree-Kirchhoff indices asymptotically obey normal distributions. We use the same notation as those used at Section 2.
Theorem 8. Suppose Hypotheses 1 and 2 are correct, then we can get the following results: (i)The variance of the Gutman index is denoted bywhere (ii)For , asymptotically obeys normal distributions. One has
Proof of Theorem 8. Let . Then by Equation (17), we have
Recalling that and are random variables which stand for our choice to construct by , we have the next two facts.
Fact 9.
Proof. If , the result is obvious. Then, we only take into account , which implies . In this case, (of ) overlaps with or (of ) (see Figure 3). In this situation, by using Equation (10) and Equation (11),
Thus, we conclude the desired fact.
Fact 10.
Similar to the proof of Fact 9, we only consider the fact , that is, . In the same way, we omit the details.
Noting that , by the above discussions, it holds that where for each ,
Therefore, by Equation (75), it follows that
By direct calculation, we have where for any two stochastic variates and , (refer to ref. [48]); using the properties of variance, Equation (79), and exchanging the order of and , we can directly find out that
By using a computer, the above expression indicates the result in Theorem 8 (i).
Now, we will go on to the proof of Theorem 8 (ii). Firstly, for any , let
By these notations, obviously, we have Then, and for some ,
Note that
By Taylor’s formula and Equations (79)–(85), one holds that
Assume that is a complex number with . We use instead of , and one has
According to the above formula ([49], Chapter 1) and the theory of continuity of probability characteristic functions ([50], Chapter 15), we complete the proof of Theorem 8 (ii).
Theorem 11. Suppose Hypotheses 1 and 2 are true, then the next results hold. (i)The variance of the Schultz index is denoted bywhere (ii)For , asymptotically obeys normal distributions. One has
Proof of Theorem 11. Let . Then by Equation (33), we get
According to the previous Proof of Theorem 8, the two facts are obtained.
Fact 12.
Proof. If , the above result is distinct. So we take into account , which indicates . In this case, (of ) coincides with or (of ) (see Figure 3). In this situation, by using Equation (10) and Equation (11), becomes
Thus, we conclude the desired Fact.
Fact 13.
As that in the proof of Fact 12, we only consider the fact , that is, . The proof is similar and details are omitted.
Noting that , by the above discussions, it holds that where for each ,
Therefore, by Equation (92),
Suppose that
If we substitute by in the Proof of Theorem 8, the rest of the proof is identical to the Proof of Theorem 8 and the details are omitted here.
Theorem 14. Suppose Hypotheses 1 and 2 are true, there are the next main results. (i)The variance of is denoted bywhere (ii)For , asymptotically obeys normal distributions. One has
Proof of Theorem 14. Obtained by formula (50), we see Let ; we have
Recalling that and are random variables which show the way to construct from , we get the following two facts.
Fact 15.
Proof. If , the result is obvious. So we just take into account , which implies . In this case, (of ) overlaps with or (of ) (see Figure 3). In this situation, by using Equation (43) and Equation (44), becomes
Thus, we conclude the desired fact.
Fact 16.
As that in the proof of Fact 15, we only consider the fact , that is, . The proof is similar and details are omitted.
Noting that , by the above discussions, it holds that where for each ,
Therefore, by Equation (102),
Through direct calculation, we have
If we substitute by in the Proof of Theorem 8, the rest of the proof is identical to the Proof of Theorem 8 and the details are omitted here.
We proceed by showing the following result about the expatiatory formula of the variance of .
Theorem 17. Suppose Hypotheses 1 and 2 are true, we can get the following two results: (i)The variance of is denoted bywhere (ii)For , asymptotically obeys normal distributions. One has
Proof of Theorem 17. Obtained by formula (64) yields, one sees that Let . Then, we obtain
Recalling that and are random variables, we obtain the following facts.
Fact 18.
Proof. If , the result is distinct. Then, we just take into account , which indicates . In this fact, (of ) coincides with or (of ) (see Figure 3). In this situation, becomes
Thus, we obtain the desired fact.
Fact 19.
Similar to the proof of Fact 18, we only consider the fact , that is, . In the same way, we omit the details.
Noting that , by the above discussions, it holds that where for each ,
Therefore, by Equation (112),
Similarly, by calculation, we can obtain the following:
The rest of the proof is similar to the above theorems, and the details are omitted here.
Remark 20. Through the above calculation, it can be found that in the random pentagonal chain networks, when , the four topological indices all obey the normal distribution; that is, combined with the expected values obtained in the second part, we can obtain their probability density functions, which have great influence in many aspects of statistics and are also very important for the application of complex networks in the fields of mathematics, physics, and engineering.
4. Conclusion
In this paper, we obtained the expected values and variances of the Gutman index, Schultz index, multiplicative degree-Kirchhoff index, and additive degree-Kirchhoff index about a class of the random pentagonal chain networks. It was calculated and observed that under the same conditions, the expected value of the Gutman index was the largest and that of multiplicative degree-Kirchhoff index was the lowest. Meanwhile, we found that for the networks , these indices all approximately obeyed the normal distribution.
The conclusion of this paper expands the research field of graph parameters related to vertex distance and degree and connects graph theory, topological indices, and statistical properties, which can more intuitively and accurately study the relevant characteristics of complex networks in various fields. In the field of chemistry, these results can be used as the basis for the study of the properties, functions, and structures of chemical molecules. In transportation, they can be used as the theoretical basis for route selection and positioning. In the field of computer applications, it can be assisted in solving problems such as network transportation and algorithm design. In a word, the research of this paper can bring some convenience to our life and study.
In the future, we will not only focus on random chain networks but also combine the complex networks in some areas of real life, take the results of this paper as the basis, and use more diverse indicators and algorithms to study and solve some practical problems.
Data Availability
No data were used to support this study.
Disclosure
An online preprint of this study has been published elsewhere [51].
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
This work was supported in part by the Anhui Provincial Natural Science Foundation under Grant 2008085J01 and by the Natural Science Fund of Education Department of Anhui Province under Grant KJ2020A0478.