# Characterization of Extremal Unicyclic Graphs UsingF-Coindex

Authors: Muhammad Imran; Yasir Ali; Zainab Bibi; Muhammad Saad Ghafar
Journal: Complexity
 (2021)

---

## 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.


---

## Body

## 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 π-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 M1 and M2, respectively. A detailed study of these indices can be found in [5]. Other than M1 and M2, 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 M1, 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, M1, and F. They came out with result that correlation coefficient between logP and F-coindex is stronger than that of M1 and F. They concluded that F-coindex can predict the logP 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 thatGVG,EG be a graph where VG and EG, respectively, denote vertex and edge sets. Also, GVG,EG is a connected graph with order VG and size EG. An adjacent edge is denoted by e=uv∈EG, whereas uw∉EG is used to denote that u and w are not adjacent. For any u∈VGoru∈G, du represents the degree of u. Here we use Nu to denote the number of vertices not adjacent to u∈G; mathematically,(1)Nu=ω|uω∉EG,for eachu∈G.Forgotten topological index is defined by Gutman and Trinajstic [4].(2)FG=∑v∈VGdv3.Equation (2) can be written as [4](3)FG=∑uv∈EGdu2+dv2.Hu et Al. [17] proposed a general form of zeroth-order Randic´ index given as follows:(4)R0αG=∑u∈Gduα=∑uv∈EGduα−1+dvα−1,αis an arbitrary real number.It can be noted that forgotten index can be obtained forα=3. Similarly, Mansour and Song [18] introduced a general formula for zeroth-order Randic´ coindex as given below:(5)R0α¯G=∑uv∉EGduα−1+dvα−1.Forα=3, the above formula takes the form of F-coindex used by De et al. [9].(6)F¯G=∑uv∉EGdu2+dv2.Very recently, Milovanovi ć et al. [19] proved the following formula for zeroth-order Randic´ coindex:(7)R0α¯G=∑u∈Gn−1−dvdvα−1,for all real values of α. Working independantly, Ali et al. [20] published the following forms of F-coindex given by equation (6):(8)F¯G=∑u∈Gdu2×Nu.Also,(9)F¯G=∑u∈Gn−1du2−du3,where the number of nonadjacent vertices to u is given by(10)Nu=n−1−du.TheF-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 Pn, Cn, K1,n−1, and Kn be the path, cycle, star, and complete graphs on n vertices. Then, F-coindex for these graphs is as follows:(1)
F¯Pn=4n2−18n+20.(2)
F¯Cn=4n2−12n.(3)
F¯K1,n−1=n−1n−2.(4)
F¯Kn=0 and F¯Kn¯=0.Note here that we reserveF¯ui to represent contribution of i vertices towards F-coindex of graph F¯G.


## 2.F-Coindex of Unicyclic Graphs

Unicyclic graphs are connected graphs with equal number of vertices and edges. LetUn denote the set of the unicyclic graphs with n vertices and Unk denote the class of all unicyclic graphs with n vertices having a cycle of length k. Let Unkp1,p2,…,pk∈Unk denote a unicyclic graph with n vertices having cycle of length k, and each vertex i of cycle has pi pendant vertices on it, where 1≤i≤k.For example,U1435,4,2, 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 Unkp1,p2,…,pk,k+p1+p2+…+pk=n.Figure 1
ForU1635,4,2.Theorem 1.
LetG∈Un and du be the degree of vertex u∈VG. Let u∗∈VG be the vertex with maximum contribution towards F¯G; then, degree of the vertex u∗ is given by(11)du∗=23n−1,ifn≡2mod3,23n−1,otherwise.Proof.
From equation (9), we know that F-coindex of a graph is given by(12)F¯G=∑u∈Gn−1du2−du3.
Letu∗ with degree du∗ have maximum contribution towards F¯; then,(13)∂F¯∂du∗=2n−1du∗−3du∗2=0,this givesdu∗=2n−13.
Also, note that(14)∂2F¯∂du∗2=−2n−1<0.
This implies that the vertex with degreedu∗=2n−1/3 will have maximum contribution towards F¯G. Note that du∗ must be an integer value. To make du∗ an integer value, we round off du∗=2n−1/3 to nearest integer value to get(15)du∗=23n−1,ifn≡2mod3,23n−1,otherwise.According to Theorem1, 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.
Forn≥4, Un3n−3,0,0 has minimum F-coindex among the family of unicyclic graphs. The F-coindex for Un3n−3,0,0 is given by(16)F¯Un3n−3,0,0=n−3n+6.Proof.
First, we will calculateF¯Un3n−3,0,0 and then show that it is minimum F-coindex among the family of unicyclic graphs.
It is obvious from the structure ofUn3n−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. The details of Un3n−3,0,0 are given in Table 1.
Using the formula given by equation (8) and information given in Table 1, we get(17)F¯Un3n−3,0,0=n−312⋅n−2+222⋅n−3=n−3n−2+8n−3=n−3n+6.
Thus,F¯Un3n−3,0,0=n−3n+6.
Now we will show thatF¯Un3n−3,0,0=n−3n+6 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(18)du∗=23n−1,ifn≡2mod3,23n−1,otherwise,as 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#1
More pendants and two-degree vertices will yield minimum F-coindex.Fact#2
Vertex with degreen−1 will have zero contribution towards F-coindex of unicyclic graph.
It is also known that total degree of a graph is2n. 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. Thus, 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. The only possible structure for such a unicyclic graph is n−3 pendants and two vertices of two-degree, that is, Un3n−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 Un3n−3,0,0.Figure 2
ForUn3n−3,0,0.Table 1
ForUn3n−3,0,0.
duNuFrequency1n−2n−32n−32n−101Theorem 3.
LetUn3p1,p2,0∈Un with p1, p2 be the pendants on u1 and u2, respectively. If p1−p2≤1, then the contribution of u1 and u2 towards F¯G is given by(19)F¯ui=14n+12n−3,ifnis odd,14n3−n2−6n−8,ifnis even,fori=1,2.Also,F¯ui, for i=1,2, is maximum when p1−p2≤1.Proof.
LetUn3p1,p2,0∈Un and n be an odd number and let p1=p2=n−3/2 be the pendants on u1 and u2, respectively. In this case, both u1 and u2 will have degree 1/2n+1 and both will be nonadjacent to 1/2n−3. Thus, contribution of both of u1 and u2 towards F-coindex is given by(20)F¯ui=2⋅12n+12n−32=14n+12n−3,fori=1,2.
Now let us assume thatUn3p1,p2,0∈Un and let n be an even number. Here u1 and u2 have p1=1/2n−2 and p2=1/2n−4, respectively. In this case, Nu1=n−4/2 and Nu2=n−2/2. Thus, we obtain F¯ui=1/4n3−n2−6n−8, for i=1,2.
Now we proveF¯ui, for i=1,2, is maximum when p1−p2≤1. On the contrary, assume that, for p1−p2>1, contribution of u∗ and u∗∗ towards F¯G is more than F¯ui for i=1,2. This means that there exists a positive integer m≥1 such that pendant at vertex u∗ is p1=n−3/2+m and similarly for u∗∗, p2=n−3/2−m and(21)F¯u∗,u∗∗>F¯ui,fori=1,2.
Initially, we assume thatn is odd. It is easy to see that(22)du∗=n+12+m,du∗∗=n+12−m.
Note that(23)Nu∗=n+12−m,Nu∗∗=n+12+m.
Thus,(24)F¯u∗,u∗∗=n+12+m2n+12−m+n+12−m2n+12+m=14n+12n−3−nm2+n2m+nm+5m2<14n+12n−3,sincem,n≥1.
Now assume thatn is even; then, we have(25)du∗=n+22+m,du∗∗=n2−m.
Note that(26)Nu∗=n−42−m,Nu∗∗=n−22+m,which gives(27)F¯u∗,u∗∗=n+22+m2n−42−m+n2−m2n+22+m=14n3−n2−6n−8−mm+1n+5<14n3−n2−6n−8,sincem,n≥1.
It is obvious that our assumption that equation (21) holds is wrong. Therefore, F¯ui, for i=1,2, is maximum when p1−p2≤1.It is important to note here that if one vertex has degreedu∗ given by (11), then maximum possible degree of any other vertex, say u∗∗, in a unicyclic graph would be n−3−du∗. Theorem 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 Theorem 3.Theorem 4.
LetUnkp1,p2,…,pk∈Unk such that pi−pj≤1, for 1≤i,j≤k; then, k=2n/n−4 maximizes F-coindex of unicyclic graph.Proof.
Without loss of generality, we assumen such that each vertex has equal vertices, that is, 1/kn−k. From Figure 3, we can have the following table.
It is obvious from the structure ofUnkp1,p2,…,pk that there would be n−k vertices of degree 1 and k vertices of degree 1/kn+k as shown in Figure 3.
Using the formula given by equation (8) and data given in Table 2, we get(28)F¯Unkp1,p2,…,pk=kn−kk+22nk−1−2kk+12n−kn−2=1k2k3n−2+k22n2−5n+kn3−4n2−n3+n2−kn−2n+2k=1kn3−4n2−n3k2+3n2−7n.
In order to get the value of that maximizesF¯Unkp1,p2,…,pk, we proceed as follows.(29)dF¯dk=0,implies thatk=2nn−4,
Also, note that fork=2n/n−4, we have d2F¯/dk2<0. This means that k=2n/n−4 maximizes F¯Unkp1,p2,…,pk.Figure 3
ForUnkp1,p2,…,pk with pi=1/kn−k, for 1≤i≤k.Table 2
ForUnkp1,p2,…,pk with pi=1/kn−k, for 1≤i≤k.
duNuFrequency1n−2n−kn−k/k+2nk−1−2k/kkWe know thatk 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.Theorem 5.
Forn≥28, let Un3p1,p2,0∈Unk, with p1,p2∈⌊n−3/2⌋,⌈n−3/2⌉, such that p1+p2=n−3. In this case,(30)F¯Un3p1,p2,0=14n−3n2+6n+9,ifnis odd,14n3+3n2−10n−32,ifnis even.Furthermore, values given by equation (30) are maximum F-coindex for the family of unicyclic graphs with n≥28.Proof.
LetUn3p1,p2,0∈Unk, with p1,p2∈ℕ such that p1+p2=n−3 and(31)p1−p2=1,ifnis even,0,ifnis odd,forn≥28.
The structure of such graphs is shown in Figure4.
From Figure4, Tables 3 and 4 are tabulated. In this case, F-coindex is obtained by adding to n−3n+2 the contribution of pendants and degree 2 vertices in equation (19), respectively for odd and even values of n. Thus using the table given above, we may write(32)F¯Un3p1,p2,0=14n−3n2+6n+9,ifnis odd,14n3+3n2−10n−32,ifnis even.
It can easily be verified thatF¯Un3p1,p2,0 given by equation (32) is maximum by using Theorems 3 and 4.Figure 4
Un3p1,p2,0, n≥28. (a) n is odd. (b) n is even.
(a)(b)Table 3
ForUn3n−3/2,n−3/2,0.
duNuFrequency1n−2n−32n−311/2n+11/2n−32Table 4
ForUn3n−2/2,n−4/2,0.
duNuFrequency1n−2n−32n−31n/21/2n−211/2n+21/2n−41Theorem 6.
For11≤n≤27, let Un3p1,p2,p3∈Unk with pi∈⌊n−3/3⌋,⌈n−3/3⌉, for i=1,2,3, such that ∑i=13pi=n−3; then,(33)F¯Un3p1,p2,p3=2n39+5n23−7n,ifn≡0mod3,6n327+5n23−53n9−4,ifn≡1mod3,2n39+53n227−7n−229,ifn≡2mod3.Proof.
For11≤n≤27, let Un3p1,p2,p3∈Unk with p1,p2,p3∈ℕ such that p1+p2+p3=n−3; then, we have following three possible cases for p1,p2, and p3.Case I:
whenn≡0mod3, then p1=p2=p3=n−3/3.Case II:
whenn≡1mod3, then p1=⌈n−3/3⌉ and p2=p3=⌊n−3/3⌋.Case III:
whenn≡2mod3, then p1=p2=⌈n−3/3⌉ and p3=⌊n−3/3⌋.
Now we consider each case separately, to find the F-coindex ofUn3p1,p2,p3 for 11≤n≤27.Case I:
whenn≡0mod3, then p1=p2=p3=n−3/3, as shown in Figure 5(a). In this class, we discuss all graphs for 11≤n≤27 when n is divisible by 3, that is, n=3k, for k=4,5,…,9. For such graphs, degree of each vertex of cycle is given by n+3/3 and there are n−3 pendant vertices. From the figure, we have the following table.Using the formula given by equation (8) and Table 5, we get(34)F¯G=3⋅n+33223n−3+n−2n−3.Simplification of the above equation gives(35)F¯G=2n39+5n23−7n.Case II:
whenn≡1mod3, then p1=n−1/3 and p2=p3=n−4/3, for n=3k+1 where k=4,5,…,8, 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(36)F¯G=2n+2322n−53+n+5322n−83+n−3n−2.After simplifying last equation, we obtain(37)F¯G=6n327+5n23−53n9−4.Case III:
in the last case, we consider the family of unicyclic graphsUn3p1,p2,p3∈Unk with n≡2mod3. Here p1=p2=n−2/3 and p3=n−5/3 where n=3k+2 for k=4,5,…,8 (Figure 5(c)). It is easy to construct the following table.Using the formula given by equation (8) and Table 7, we get(38)F¯G=2n+4322n−73+n+1322n−43+n−3n−2.Thus, we obtain(39)F¯G=2n39+53n227−7n−229.Table 5
ForUn3p1,p2,p3.
duNuFrequencyn+3/32/3n−331n−2n−3Table 6
ForUn3p1,p2,p3.
duNuFrequency1/3n+21/32n−521/3n+52/3n−411n−2n−3Table 7
ForUn3p1,p2,p3.
duNuFrequency1/3n+41/32n−721/3n+12/3n−211n−2n−3Figure 4
Un3p1,p2,p3 with pi∈⌊n−3/3⌋,⌈n−3/3⌉, for i=1,2,3.
(a)(b)(c)Theorem 7.
LetUn3p1,p2,p3∈Unk with pi∈⌊n−3/3⌋,⌈n−3/3⌉, for i=1,2,3, such that p1+p2+p3=n−3; then, F¯Un3p1,p2,p3 has maximum F-coindex for the family of unicyclic graphs with 11≤n≤27.Proof.
By Theorem4, we know that k=2n/n−4 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 k given in Theorem 4 is close to 3 when n=11 and it decreases as we increase n. Therefore, we can compare F-coindex of both values of k, that is, k=2 and k=3.
Without loss of generality, we assume thatn−3 is even and divisible by 3, and it is equivalent to saying that n is odd and divisible by 3. It is evident from Figure 6 and Tables 8 and 9 that both Un3p1,p2,p3 and Un3p1,p2,0 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 forF-coindex of Un3p1,p2,p3 and Un3p1,p2,0. We can see that in Un3p1,p2,p3, there are 3 vertices, say u1,u2, and u3, with degree n+3/3 with nonadjacent vertices 2n−3/3.(40)F¯ui=29n+32n−3,fori=1,2,3,where F¯ui, for i=1,2,3, represents contribution of u1,u2, and u3 towards F¯Un3n−3/3,n−3/3,n−3/3.
Similarly, forUn3p1,p2,0, contribution of v1,v2, and v3 is given by(41)F¯vi=14n−3n2+2n+17,fori=1,2,3,where F¯vi, for i=1,2,3, represents contribution of v1, v2, and v3 towards F¯Un3n−3/2,n−3/2,0. By analyzing equations (40) and (41), we have(42)F¯ui=F¯vi,forn=27,andF¯ui>F¯vifor11≤n<27.
In general, we can write(43)F¯Un3p1,p2,p3>F¯Un3p1,p2,0,for11≤n<27,F¯Un3p1,p2,p3=F¯Un3p1,p2,0,forn=27,F¯Un3p1,p2,p3<F¯Un3p1,p2,0,forn>27.
Therefore, for11≤n≤27, F¯Un3p1,p2,p3 has maximum F-coindex for the family of unicyclic graphs.Figure 4
Un3p1,p2,p3 and Un3p1,p2,0.
(a)(b)Table 8
ForUn3n−3/3,n−3/3,n−3/3.
duNuFrequencyn+3/32/3n−331n−2n−3Table 9
ForUn3n−3/2,n−3/2,0.
duNuFrequency1n−2n−32n−31n+1/2n−3/22


## 3. Conclusions

In this paper, we introduced an alternative form ofF-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 studyF-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.



---

*Source: 8480971-2021-08-26.xml*