Abstract
Given , a hexagonal chain, we determine an expression of its second-order general connectivity index, denoted by , in terms of inlet features of . Moreover, by applying the method in integer programming theory, we completely determine the extremal chains with the minimal or maximal over the set of hexagonal chains.
1. Introduction
A hexagonal system is a finite connected plane graph without cut vertices, in which every interior face is bounded by a regular hexagon of side of length one. A hexagonal system with hexagons is called an -hexagonal system for short. Hexagonal systems are of great importance for theoretical chemistry because they are the natural graph representation of benzenoid hydrocarbons. A considerable amount of research in mathematical chemistry has been devoted to hexagonal systems [1–5].
The following definitions were introduced in [6, 7]. Suppose is a hexagonal system. We can associate to each path of length in , with the vertex degree sequence . If one goes along the perimeter of , then a fissure, bay, cove, and fjord are, respectively, paths of degree sequences , , , and (see Figure 1(i)). The fissures, bays, coves, and fjords are called various types of inlets. The number of inlets of is defined as the sum of the numbers of fissures, bays, coves, and fjords.

A hexagonal system without internal vertex is called a catacondensed (hexagonal) system. A hexagonal system is called a hexagonal chain if it has no hexagon adjacent to more than two hexagons. Denote by the set of vertices of degree in . For an -hexagonal chain , its induced subgraph is an acyclic graph. Moreover, if the subgraph is matching with edges, then is called a linear (hexagonal) chain and denoted by . If the subgraph is a path, then is called a zig-zag chain and denoted by . Some examples of hexagonal chains can be found in Figure 1(ii), where are indicated by heavy edges. For a simple graph and an interger , the th-order general connectivity index of is defined aswhere the sum goes over all paths of length in (we do not distinguish between the paths and ), is the degree of vertex , and is a real number not equal to .
In general, for , we call the higher-order general connectivity index of .
The th-order general connectivity index is a natural generalization of some topological indices in mathematical chemistry. It is well known that , called the connectivity index of , was introduced by Randić in 1975 in the study of branching properties of alkanes. is called the th-order connectivity index of and was considered in 1976 by Randić, Kier, Hall, and coworkers [8] with the intention of extending the applicability of the connectivity index. Moreover, is the general connectivity index (also called the general Randić index) of , which was studied in 1998 by Bollobás and Erdös [9].
Experimental results show that these indices are closely correlated with many physical, chemical, and biological properties and found to parallel the boiling point, solubility, toxicity, Kovats constants, and a calculated surface. For the results related to the mathematical properties of these indices, please refer to the literature [10].
There are many contributions on the mathematical properties of these indices for hexagonal systems. In [11], the authors proved that is completely determined by the numbers of hexagons and inlets of . Hexagonal systems with extremal have also been found [6]. Rada [7] gave an expression of and found the extremal value over the set of catacondensed systems. For a catacondensed system , Zheng [12] gave a formula for computing , and the catacondensed systems with the first up to the third extremal value of were completely characterized. For other results, please refer to [13–17].
In this work, we are interested in the second-order general connectivity indices of hexagonal chains. It is shown in Theorem 1 that, for a hexagonal chain , is completely determined by the numbers of hexagons, bays, coves, and fjords of . The expression for is applied in Theorem 4 to obtain the extremal hexagonal chains with the minimal or maximal among all hexagonal chains with hexagons.
2. A Formula for Computing
Theorem 1. For an -hexagonal chain with bays , coves, and fjords, its second-order general connectivity index is equal to
Proof. We will prove this theorem by mathematical induction method. If , then is a linear chain with hexagons. Its second-order general connectivity index is . So (2) is true for .
Now we apply induction on . Assume is a hexagonal chain with bays, coves, and fjords and (2) holds for any hexagonal chain with bays, coves, and fjords. Suppose is a fjord in . Then and . Let be the vertex so that is an edge of and , . Let be the vertex so that is an edge of and , . Cut into three parts , and a single hexagon along the edges and , respectively. See Figure 2.
Suppose has hexagons, bays, coves, and fjords and has hexagons, bays, coves, and fjords. Then , , , and . Using the inductive hypothesis, and Note that when we cut into , and a single hexagon following the above method, only the paths in which contain at least one element in have been changed. Comparing , , and , we have So (2) is true for with bays, coves, and fjords.
Similarly, we can apply inductions on and , respectively. Then the proof for the theorem is completed.

3. Extremal Problem
As mentioned in [11], for an -hexagonal system with bays, coves, and fjords, the number of bay regions in is . Furthermore, if is an -hexagonal chain, then each hexagon in corresponds to at most one bay region and the two hexagons which are the ends of correspond to no bay region. So the following lemma is obvious.
Lemma 2. Suppose is an -hexagonal chain with bays, coves, and fjords. Then .
The next lemma plays a significant role in the proof of the main results in this section.
Lemma 3. Suppose is an -hexagonal chain with bays, coves, and fjords. Let and , , and . Then, for the function , we have (1) for (2) and , where is the negative root of the equation
Proof. Firstly, if , then it is easy to get and . By Lemma 2, . Then . The equalities hold if and only if and .
Secondly, if , then . We next apply the methods in the theory of linear programming and integer programming. (For the notations and terminology not introduced here, please refer to [18] for more details.) Consider a linear integer programming problem (IP) and its corresponding linear programming relaxation problem (LR). By the simplex method, we know the optimal solution for (LR) is with .
By the branch-and-bound method, if , then the optimal solution for (LR) is exactly the optimal solution for (IP). Then for (IP), .
If , then by applying the branch-and-bound method, we obtain that the optimal solution for (IP) is with . The flow diagram is shown in Figure 3.
If , then, by the branch-and-bound method, the optimal solution for (IP) is with . The flow diagram is shown in Figure 4. Then we prove


Theorem 4. For , the extremal -hexagonal chains with minimal second-order general connectivity index are shown in Table 1.
Proof. Let and be the linear hexagonal chain and the zig-zag chain with hexagons, respectively. Obviously, has no bay, cove, or fjord and only has bays, coves, and fjords. By Theorem 1,Suppose is an -hexagonal chain with bays, coves, and fjords. Then, by Lemma 2 and Theorem 1, we have , , , and Comparing and , we have If , then , , . Thus we have . The equality holds if and only if (). Then we prove that is the unique extremal hexagonal chain with minimal when .
Comparing and , we have By Lemma 3, if , . The equality holds if and only if and . Then we prove that is the unique extremal hexagonal chain with minimal when .
Let be the set of -hexagonal chains so that if and only if is a hexagonal chain with fjords, coves, and bays. Obviously, and .
Theorem 5. For , the extremal -hexagonal chains with maximal second-order general connectivity index are shown in Table 2, where and are the negative roots of the equations and , respectively.
Proof. We here only prove the results for the case that . Other cases are completely similar. Suppose is an -hexagonal chain with bays, coves, and fjords. By Theorem 1, we have . As defined in Lemma 3, , , and denote , , and , respectively.(1)If and , then has bays, cove, and fjords. By Theorem 1, . Then, by Lemma 3, Thus we prove that .(2)If and , then has bays, coves, and fjords. By Theorem 1, . Then, by Lemma 3, Thus we prove that .(3)If and , then has bays, coves, and fjords. By Theorem 1, . Then, by Lemma 3, Thus we prove that .By the above analysis, we have the following result. For , if , then the hexagonal chains in have maximal ; if , then the hexagonal chains in have maximal .
The discussion for other cases is similar and we omit it here.
Example 6. Table 3 shows the second-order general connectivity indices of , , and some hexagonal chains in Figure 5. The extremal values are indicated by heavy words.

Data Availability
No additional data are available.
Conflicts of Interest
The author declares that they have no conflicts of interest.
Acknowledgments
This research is supported by the East China University of Political Science and Law’s research projects (A-0333-18-139015) and Beijing Qihoo Science and Technology Co., Ltd. cooperation project (QH-E-201803-00200).