Characterization of Extremal Unicyclic Graphs Using F -Coindex

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.


Introduction
ere 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 highquality 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][2][3]). In 1972, Gutman and Trinajstic [4] approximated formulas for the total π-electron energy (E). Among the formulas presented in [4], a pair of topological indices was studied more frequently than any other indices. ese pairs of indices are known as Zagreb first and second indices denoted by symbols M 1 and M 2 , respectively. A detailed study of these indices can be found in [5]. Other than M 1 and M 2 , 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 M 1 , 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 (P) with F-coindex, M 1 , and F. ey came out with result that correlation coefficient between log P and Fcoindex is stronger than that of M 1 and F. ey concluded that F-coindex can predict the log P values with high accuracy. e 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][11][12][13][14]. Recently, Amin et al. [15] studied about minimal trees using F-coindex. e 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. roughout the work we assume that G(V(G), E(G)) be a graph where V(G) and E(G), respectively, denote vertex and edge sets. Also, G(V(G), E(G)) is a connected graph with order |V(G)| and size |E(G)|. An adjacent edge is denoted by e � uv ∈ E(G), whereas uw ∉ E(G) is used to denote that u and w are not adjacent. For any u ∈ V(G)(or u ∈ G), d u represents the degree of u. Here we use N u to denote the number of vertices not adjacent to u ∈ G; 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 Randic � index given as follows: α is an arbitrary real number.

(4)
It can be noted that forgotten index can be obtained for α � 3. Similarly, Mansour and Song [18] introduced a general formula for zeroth-order Randic � coindex as given below: For α � 3, the above formula takes the form of F-coindex used by De et al. [9].
Very recently, Milovanović et al. [19] proved the following formula for zeroth-order Randic � coindex: for all real values of α. 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 u is given by e 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 P n , C n , K 1,n− 1 , and K n be the path, cycle, star, and complete graphs on n vertices. en, Fcoindex for these graphs is as follows: Note here that we reserve F u i to represent contribution of i vertices towards F-coindex of graph F(G).

F-Coindex of Unicyclic Graphs
Unicyclic graphs are connected graphs with equal number of vertices and edges. Let U n denote the set of the unicyclic graphs with n vertices and U k n denote the class of all unicyclic graphs with n vertices having a cycle of length k. Let U k n (p 1 , p 2 , . . . , p k ) ∈ U k n denote a unicyclic graph with n vertices having cycle of length k, and each vertex i of cycle has p i pendant vertices on it, where 1 ≤ i ≤ k.
For example, U 3 14 (5, 4, 2), 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 G ∈ U n and d u be the degree of vertex u ∈ V(G). Let u * ∈ V(G) be the vertex with maximum contribution towards F(G); then, degree of the vertex u * is given by Proof. From equation (9), we know that F-coindex of a graph is given by Let u * with degree d u * have maximum contribution towards F; then, Also, note that is implies that the vertex with degree d u * � (2(n − 1)/3) will have maximum contribution towards F(G). Note 2 Complexity that d u * must be an integer value. To make d u * an integer value, we round off d u * � (2(n − 1)/3) to nearest integer value to get □ According to eorem 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).
It is obvious from the structure of U 3 n (n − 3, 0, 0) 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 n − 1 having n − 3 pendants. e details of U 3 n (n − 3, 0, 0) are given in Table 1.
Using the formula given by equation (8) and information given in Table 1, we get 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 until as given in eorem 1. We can note from eorem 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. is can be further categorized as the following two facts.
Fact#1 More pendants and two-degree vertices will yield minimum F-coindex. Fact#2 Vertex with degree n − 1 will have zero contribution towards F-coindex of unicyclic graph.
It is also known that total degree of a graph is 2n. If one of the vertices has degree n − 1 with zero contribution towards F-coindex of unicyclic graph, then remaining n − 1 vertices will have total degree n + 1. us, instead of total degree 2n, we are left with the degree n + 1 that is contributing towards F-coindex of unicyclic graph in the form of pendants and two-degree vertices. e only possible structure for such a unicyclic graph is n − 3 pendants and two vertices of twodegree, that is, U 3 n (n − 3, 0, 0). 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 U 3 n (n − 3, 0, 0).

Complexity 3
Theorem 3. Let U 3 n (p 1 , p 2 , 0) ∈ U n with p 1 , p 2 be the pendants on u 1 and u 2 , respectively. If |p 1 − p 2 | ≤ 1, then the contribution of u 1 and u 2 towards F(G) is given by if n is odd, Also, F u i , for i � 1, 2, is maximum when |p 1 − p 2 | ≤ 1.

(24)
Now assume that n is even; then, we have Note that which gives It is obvious that our assumption that equation (21) holds is wrong. erefore, F u i , for i � 1, 2, is maximum when |p 1 − p 2 | ≤ 1.

□
It is important to note here that if one vertex has degree d u * given by (11), then maximum possible degree of any 4 Complexity other vertex, say u * * , in a unicyclic graph would be n − 3 − d u * . eorem 3 says that the contribution of both u * and u * * towards F(G) would be less than that if both vertices having degrees states in eorem 3.
Proof. Without loss of generality, we assume n such that each vertex has equal vertices, that is, (1/k)(n − k). From Figure 3, we can have the following table.
It is obvious from the structure of U k n (p 1 , p 2 , . . . , p k ) that there would be (n − k) vertices of degree 1 and k vertices of degree (1/k)(n + k) as shown in Figure 3.
Using the formula given by equation (8) and data given in Table 2, we get In order to get the value of that maximizes F(U k n (p 1 , p 2 , . . . , p k )), we proceed as follows.

□
We know that k must be a positive integer. From the expression k � (2n/(n − 4)), it can be seen that for large values of n, k approaches 2 but for values of n close to 11, k is closer to 3.
Proof. Let U 3 n (p 1 , p 2 , 0) ∈ U k n , with p 1 , p 2 ∈ N such that p 1 + p 2 � n − 3 and e 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 (n − 3)(n + 2) the contribution of pendants and degree 2 vertices in equation (19), respectively for odd and even values of n. us using the table given above, we may write if n is odd, It can easily be verified that F(U 3 n (p 1 , p 2 , 0)) given by equation (32) is maximum by using eorems 3 and 4. □ Proof. For 11 ≤ n ≤ 27, let U 3 n (p 1 , p 2 , p 3 ) ∈ U k n with p 1 , p 2 , p 3 ∈ N such that p 1 + p 2 + p 3 � n − 3; then, we have following three possible cases for p 1 , p 2 , and p 3 .
Using the formula given by equation (8) and Table 6, we get
Proof. By eorem 4, we know that k � (2n/(n − 4)) maximizes F-coindex of unicyclic graph. is value must be integer as it represents number of vertices on which we should divide pendant vertices. e value of k given in eorem 4 is close to 3 when n � 11 and it decreases as we increase n. erefore, we can compare F-coindex of both values of k, that is, k � 2 and k � 3.

Conclusions
In this paper, we introduced an alternative form of Fcoindex 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.
is 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
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.