Abstract
The study of forgotten index and coindex for the molecular structures of some special chemical graphs (compounds and drugs) has proven significant in medical and pharmaceutical drug design fields by making reliable statistical conclusion about biological properties of new chemical compounds and drugs. In mathematical chemistry, finding extreme graphs with respect to topological index is an active area of research. The aim of this paper is to characterize the family of unicyclic graphs with extreme (largest and smallest) F-coindex. Moreover, the study also contains different other properties of unicyclic graphs. All these results are based upon an alternative form of F-coindex which is established with the help of special property in the graphs.
1. Introduction
There is a continuous rise in various new diseases every year, the reason being different novel viruses and bacteria. In order to overcome such diseases, introduction of high-quality drugs is of prime significance. Importance of topological indices is a well-established fact for nanotechnology in isomer discrimination, pharmaceutical drug design, structure-property relationship, and structure-activity relationship (for details, see [1–3]). In 1972, Gutman and Trinajstic [4] approximated formulas for the total -electron energy (E). Among the formulas presented in [4], a pair of topological indices was studied more frequently than any other indices. These pairs of indices are known as Zagreb first and second indices denoted by symbols and , respectively. A detailed study of these indices can be found in [5]. Other than and , another formula depending upon sum of cubes of vertex degrees was also presented in [4]. In order to shed some light on this important index, Furtula and Gutman [6] established some of its basic properties of F-index (“forgotten topological index”). Furtula and Gutman [6] further showed that F-index can improve physicochemical applicability of , significantly. Also, by assuming the linear model, they enhanced the predictive ability of these indices. While working on weighted Wiener polynomial of graphs, Doslic [7] initiated work on coindices by introducing Zagreb coindices. Formal definitions of Zagreb coindices and basic properties were reported by Ashrafi et al. [8]. De et al. [9] studied some key features of F-coindex. In [9], the authors compared the correlation of the logarithm of the octanol-water partition coefficient () with F-coindex, , and . They came out with result that correlation coefficient between and F-coindex is stronger than that of and . They concluded that F-coindex can predict the values with high accuracy. The study of F-index and F-coindex is an active area of research nowadays, and we can find a number of results, for example, [10–14]. Recently, Amin et al. [15] studied about minimal trees using F-coindex. The results in [15] are only about first, second, and third minimum trees. Very recently, Akther et al. [16] deduced the results for extremal graphs for F-index among the classes of connected unicyclic and bicyclic graphs. In the present work, we provide an alternative form of F-coindex to study different properties of unicyclic graphs. Moreover, these properties are further used to study extremal graphs for F-coindex for the family of unicyclic graphs.
Throughout the work we assume that be a graph where and , respectively, denote vertex and edge sets. Also, is a connected graph with order and size . An adjacent edge is denoted by , whereas is used to denote that and are not adjacent. For any , represents the degree of . Here we use to denote the number of vertices not adjacent to ; mathematically,
Forgotten topological index is defined by Gutman and Trinajstic [4].
Equation (2) can be written as [4]
Hu et Al. [17] proposed a general form of zeroth-order Randi index given as follows:
It can be noted that forgotten index can be obtained for . Similarly, Mansour and Song [18] introduced a general formula for zeroth-order Randi coindex as given below:
For , the above formula takes the form of F-coindex used by De et al. [9].
Very recently, Milovanovi ć et al. [19] proved the following formula for zeroth-order Randi coindex:for all real values of . Working independantly, Ali et al. [20] published the following forms of F-coindex given by equation (6):
Also,where the number of nonadjacent vertices to is given by
The F-coindex is also called Lanzhou Index. In 2021, Dehgardi and Liu [21] characterized trees with fixed maximum degree using the Lanzhou Index. We can easily calculate F-coindex, using formula given by equation (8), for some well-known graphs. Let , , , and be the path, cycle, star, and complete graphs on vertices. Then, F-coindex for these graphs is as follows:(1).(2).(3).(4) and .
Note here that we reserve to represent contribution of vertices towards F-coindex of graph .
2. F-Coindex of Unicyclic Graphs
Unicyclic graphs are connected graphs with equal number of vertices and edges. Let denote the set of the unicyclic graphs with vertices and denote the class of all unicyclic graphs with vertices having a cycle of length . Let denote a unicyclic graph with vertices having cycle of length , and each vertex of cycle has pendant vertices on it, where .
For example, , shown in Figure 1, represents a unicyclic graph with 14 vertices having a cycle of length 3, where these 3 vertices of cycle have 5, 4, and 2 pendants, respectively. It is important to note that in .
Theorem 1. Let and be the degree of vertex . Let be the vertex with maximum contribution towards ; then, degree of the vertex is given by
Proof. From equation (9), we know that F-coindex of a graph is given byLet with degree have maximum contribution towards ; then,Also, note thatThis implies that the vertex with degree will have maximum contribution towards . Note that must be an integer value. To make an integer value, we round off to nearest integer value to get
According to Theorem 1, it is evident that contribution of a vertex towards F-coindex in a unicyclic graph increases with an increase in degree of the vertex until degree of the vertex reaches a value given by equation (11).
Theorem 2. For , has minimum F-coindex among the family of unicyclic graphs. The F-coindex for is given by
Proof. First, we will calculate and then show that it is minimum F-coindex among the family of unicyclic graphs.
It is obvious from the structure of that there would be a cycle of length 3 and all pendant vertices will be there on a single vertex of the cycle and there would be two vertices of degree 2, as shown in Figure 2. From Figure 2, it is clear that one vertex of the cycle has degree having pendants. The details of are given in Table 1.
Using the formula given by equation (8) and information given in Table 1, we getThus, .
Now we will show that is the minimum value of F-coindex for the family of unicyclic graph. From equations (8) and (9), contribution towards F-coindex of unicyclic graph increases with increase in degree of the vertex untilas given in Theorem 1. We can note from Theorem 1 and equation (9) that the F-coindex can be minimized by minimizing each of the term that is contributing for each of the vertices. This can be further categorized as the following two facts.Fact#1More pendants and two-degree vertices will yield minimum F-coindex.Fact#2Vertex with degree will have zero contribution towards F-coindex of unicyclic graph.It is also known that total degree of a graph is . If one of the vertices has degree with zero contribution towards F-coindex of unicyclic graph, then remaining vertices will have total degree . Thus, instead of total degree , we are left with the degree that is contributing towards F-coindex of unicyclic graph in the form of pendants and two-degree vertices. The only possible structure for such a unicyclic graph is pendants and two vertices of two-degree, that is, . It is likewise established that in any other structure of unicyclic graph, there would be more vertices with higher degrees, ultimately contributing more towards F-coindex of unicyclic graph than .
Theorem 3. Let with , be the pendants on and , respectively. If , then the contribution of and towards is given by
Also, , for , is maximum when .
Proof. Let and be an odd number and let be the pendants on and , respectively. In this case, both and will have degree and both will be nonadjacent to . Thus, contribution of both of and towards F-coindex is given byNow let us assume that and let be an even number. Here and have and , respectively. In this case, and . Thus, we obtain , for .
Now we prove , for , is maximum when . On the contrary, assume that, for , contribution of and towards is more than for . This means that there exists a positive integer such that pendant at vertex is and similarly for , andInitially, we assume that is odd. It is easy to see thatNote thatThus,Now assume that is even; then, we haveNote thatwhich givesIt is obvious that our assumption that equation (21) holds is wrong. Therefore, , for , is maximum when .
It is important to note here that if one vertex has degree given by (11), then maximum possible degree of any other vertex, say , in a unicyclic graph would be . Theorem 3 says that the contribution of both and towards would be less than that if both vertices having degrees states in Theorem 3.
Theorem 4. Let such that , for ; then, maximizes F-coindex of unicyclic graph.
Proof. Without loss of generality, we assume such that each vertex has equal vertices, that is, . From Figure 3, we can have the following table.
It is obvious from the structure of that there would be vertices of degree 1 and vertices of degree as shown in Figure 3.
Using the formula given by equation (8) and data given in Table 2, we getIn order to get the value of that maximizes , we proceed as follows.Also, note that for , we have . This means that maximizes .
We know that must be a positive integer. From the expression , it can be seen that for large values of , approaches 2 but for values of close to 11, is closer to 3.
Theorem 5. For , let , with , such that . In this case,
Furthermore, values given by equation (30) are maximum F-coindex for the family of unicyclic graphs with .
Proof. Let , with such that andThe structure of such graphs is shown in Figure 4.
From Figure 4, Tables 3 and 4 are tabulated. In this case, F-coindex is obtained by adding to the contribution of pendants and degree 2 vertices in equation (19), respectively for odd and even values of . Thus using the table given above, we may writeIt can easily be verified that given by equation (32) is maximum by using Theorems 3 and 4.
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Theorem 6. For , let with , for , such that ; then,
Proof. For , let with such that ; then, we have following three possible cases for , and .Case I:when , then .Case II:when , then and .Case III:when , then and .Now we consider each case separately, to find the F-coindex of for .Case I:when , then , as shown in Figure 5(a). In this class, we discuss all graphs for when is divisible by 3, that is, , for . For such graphs, degree of each vertex of cycle is given by and there are pendant vertices. From the figure, we have the following table. Using the formula given by equation (8) and Table 5, we get Simplification of the above equation givesCase II:when , then and , for where , as shown in Figure 5(b). In this case, we have the following values of degrees, nonadjacent vertices, and frequencies. Using the formula given by equation (8) and Table 6, we get After simplifying last equation, we obtainCase III:in the last case, we consider the family of unicyclic graphs with . Here and where for (Figure 5(c)). It is easy to construct the following table. Using the formula given by equation (8) and Table 7, we get Thus, we obtain
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Theorem 7. Let with , for , such that ; then, has maximum F-coindex for the family of unicyclic graphs with .
Proof. By Theorem 4, we know that maximizes F-coindex of unicyclic graph. This value must be integer as it represents number of vertices on which we should divide pendant vertices. The value of given in Theorem 4 is close to 3 when and it decreases as we increase . Therefore, we can compare F-coindex of both values of , that is, and .
Without loss of generality, we assume that is even and divisible by 3, and it is equivalent to saying that is odd and divisible by 3. It is evident from Figure 6 and Tables 8 and 9 that both and have same pendant vertices with same number of nonadjacent vertices.
Therefore, contribution of pendant vertices would be same in both cases. It means that we should compare the contribution of other vertices for F-coindex of and . We can see that in , there are 3 vertices, say , and , with degree with nonadjacent vertices .where , for , represents contribution of , and towards .
Similarly, for , contribution of , and is given bywhere , for , represents contribution of , , and towards . By analyzing equations (40) and (41), we haveIn general, we can writeTherefore, for , has maximum F-coindex for the family of unicyclic graphs.
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3. Conclusions
In this paper, we introduced an alternative form of F-coindex of graphs. We used this form to study different properties of unicyclic graphs. We also calculated maximum and minimum F-coindex for the family of unicyclic graphs.
This alternative form may be used to study F-coindex of some special and chemically interesting graphs. It may help in studying F-coindex of other classes of graphs like bicyclic and tricyclic classes. Furthermore, it would be an important direction if one can explore similar formula for other topological coindices.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.