# New Results on the Forgotten Topological Index and Coindex

Authors: Akbar Jahanbani; Maryam Atapour; Rana Khoeilar
Journal: Journal of Mathematics
 (2021)

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## Abstract

Theℱ-coindex (forgotten topological coindex) for a simple connected graph G is defined as the sum of the terms ζG2y+ζG2x over all nonadjacent vertex pairs x,y of G, where ζGy and ζGx are the degrees of the vertices y and x in G, respectively. The ℱ-index of a graph is defined as the sum of cubes of the vertex degrees of the graph. This was introduced in 1972 in the same paper where the first and second Zagreb indices were introduced to study the structure dependency of total π-electron energy. Therefore, considering the importance of the ℱ-index and ℱ-coindex, in this paper, we study these indices, and we present new bounds for the ℱ-index and ℱ-coindex.


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## Body

## 1. Introduction

SupposeG be a simple graph with vertex set V=VG and edge set EG. The integers ℵ=ℵG=VG and ϵ=ϵG=EG are the order and the size of the graph G, respectively; we say G is a ℵ,ϵ-graph. The open neighborhood of vertex v is NGx=Nx=yϵVG|xyϵEG, and the degree of v is ζGx=Nx. We write Θ and δ for the maximum and minimum degrees of G, respectively. A graph G is said to be t-regular if all of its vertices have degree t. An r,s-semiregular graph is a graph whose each vertex is of degree s or r, and a k,s,t-triangular graph is a graph whose each vertex is of degree k,s, or t. The complement G¯ of a graph G is a graph that has the same vertices as G and in which two vertices are adjacent if and only if they are not adjacent in G. The number of vertex pairs xi,xj in G such that xixj∉EG is ϵ¯=ℵℵ−1/2−ϵG. A pendant vertex is a vertex of degree one. The number of pendant vertices in G is denoted by r=rG. We denote by π1 the minimal nonpendant vertex degree.In [1, 2], the first and second Zagreb indices are defined as the following:(1)M1=M1G=∑x∈VGζG2x=∑xy∈EGζGy+ζGx,M2=M2G=∑xy∈EGζGyζGx,respectively. The first and second Zagreb coindices are defined in [3, 4] as(2)M¯1G=∑xy∉EGζGy+ζGx,M¯2G=∑xy∉EGζGyζGx.Furtula and Gutman [5] defined the forgotten topological index (ℱ-index) as the following:(3)ℱ=ℱG=∑x∈VGζG3x=∑xy∈EGζG2y+ζG2x.For more information on theℱ-index, see [6–8].Theℱ-coindex introduced in [9] is as follows:(4)ℱ¯G=∑xy∉EGζG2y+ζG2x.Gutman in [10] introduced the hyper-Zagreb coindex as follows:(5)HM¯G=∑xy∉EGζGy+ζGx2.Usha et al. [11] introduced the redefined first Zagreb indices as the following:(6)ReZG1G=∑xy∈EGζGy+ζGxζGyζGx.Here, we introduce the redefined first Zagreb coindex as the following:(7)ReZG¯1G=∑xy∉EGζGy+ζGxζGyζGx.The authors introduced the first general Zagreb coindex in [12], and it is defined as follows:(8)M¯1αG=∑xy∉EGζGα−1y+ζGα−1x,where α∈R. The second general Zagreb coindex was introduced in [13] and defined as follows:(9)M¯2αG=∑xy∉EGζGyζGxα.Topological indices are numerical quantity derived from a molecular graph which correlate the physicochemical properties of the molecular graph. Recently, topological indices have been studied by many researchers due to their applications in various sciences such as chemistry, physics, and electricity; see [14–16]. Among the topological indices, the first Zagreb index is one of the oldest and most applied topological indices, and for this reason, it is of great importance and has been considered by many researchers today. Furtula and Gutman in [5] recently investigated this index and named this index as “forgotten topological index” or “F-index” and showed that the predictive ability of this index is almost similar to that of the first Zagreb index and for the entropy and acetic factor; both of them yield correlation coefficients greater than 0.95. Therefore, due to the importance of the F-index in this paper, we have decided to study this index.In [17, 18], some of bounds for the general Zagreb coindices were obtained. Ranjini et al. [19] presented some of the bounds for Zagreb indices and the Zagreb coindices. In [20], some bounds were presented for the ℱ-index and ℱ-coindex. For more other bounds, see [21–23].Given the importance of the forgotten topological index and the fact that it has recently attracted the attention of researchers and the interest of many readers, in this paper, we intend to discuss new bounds for this index.


## 2. Preliminaries

Here, we recall several published results that we will need for proof.The following result obtained the relationship between the first Zagreb index and the maximum and minimum degrees.Theorem 1 (see [24]).
LetG be an ℵ,ϵ-connected graph and ℵ≥2. Then,(10)M1≥4ϵ2ℵ+12Θ−δ2,with equality if and only if G is isomorphic with a regular graph.The following result comes from [18].Theorem 2.
SupposeG be a ℵ,ϵ-graph and ℵ≥2. Then,(11)M1≤4ϵ2ℵ+ℵ4Θ−δ2,with equality if and only if G is isomorphic with the t-regular graph, 1≤t≤ℵ−1.In [19], the authors gave the relation of the second Zagreb coindex, the maximum degree, and the first Zagreb index as follows.Theorem 3.
LetGbe anℵ,ϵ-graph and maximal degree be Θ. Then,(12)M¯2G≤2Θϵℵ−1−ΘM1G.Zhou and Trinajstić [25] proved the following result.Theorem 4.
SupposeG be a ℵ,ϵ-graph. Then,(13)ℱG≥16ϵ3ℵ2−2M2G,with equality if and only if G is regular.Furtula and Gutman [5] showed the following.Theorem 5.
SupposeG be a graph of size ϵ. Then,(14)ℱG≥M1G22ϵ,with equality if and only if G is regular.In [26], Elumalai et al. obtained the following two results.Theorem 6.
SupposeG be a simple graph of order ℵ≥2. Then,(15)ℱG≥Θ3+Θ23+Φ,with equality if and only if G is regular, where(16)Φ=M1G−Θ2+Θ222+ℵ−2M1G−Θ2+Θ222ϵ−Θ+Θ2−2ϵ−Θ+Θ2.Theorem 7.
SupposeG be a simple graph of order ℵ≥2. Then,(17)ℱG≥Θ3+Θ23+Y,with equality if and only if G is regular, where(18)Y=M1G−Θ2−δ22+2ϵ−Θ−δIDG−1/Θ−1/δ−ℵ−222ϵ−Θ−δ.In [5], Furtula and Gutman mentioned the following result.Theorem 8.
SupposeG be a connected ℵ,ϵ-graph and second Zagreb index M2G. Then,(19)ℱG≤2M2G+ϵℵ−2,with equality if and only if G is the star graph.The following result was first proved in [27].Theorem 9.
SupposeG be a simple graph of order ℵ. Then,(20)ℱ¯G+ℱG=ℵ−1M1G.


## 3. New Bounds for theℱ-Coindex

In this section, we will obtain some bounds for theℱ-coindex in terms of the maximal degree, minimum degree, order, size, pendant vertex, and the first and the second Zagreb indices.Theorem 10.
SupposeG be a ℵ,ϵ-graph. Then,(21)ℱG¯+ℱG=2ℵ−12ϵ¯−2ϵ+3ℵ−1M1G.Proof.
By applying the definition of theℱ-index for complement graphs, we have(22)ℱG¯=∑x∈VGζG¯3x=∑x∈VGℵ−1−ζGx3,=∑x∈VGℵ−13−ζG3x−3ℵ−1ζG2x+3ℵ−12ζGx,=∑x∈VGℵ−13−∑x∈VGζG3x+3ℵ−1∑x∈VGζG2x−3∑x∈VGℵ−12ζGx,=ℵℵ−13−ℱG+3ℵ−1M1G−6ϵℵ−12,=2ℵ−12ϵ¯−2ϵ−ℱG+3ℵ−1M1G.Theorem 11.
SupposeG be a ℵ,ϵ-graph. Then,(23)ℱ¯G¯=2ϵℵ−12−2ℵ−1M1G+ℱG.Proof.
For any vertexy∈VG, we have ζG¯y=ℵ−1−ζGy, and by applying the definition of the ℱ-coindex, we have(24)ℱ¯G¯=∑xy∉EG¯ζG¯2y+ζG¯2x,=∑xy∈EGℵ−1−ζGx2+ℵ−1−ζGy2,=∑xy∈EGℵ−12+ζG2x−2ℵ−1ζGx+ℵ−12+ζG2y−2ℵ−1ζGy,=∑xy∈EG2ℵ−12−2ℵ−1ζGx+ζGy+ζG2x+ζG2y,=2ℵ−12ϵ−2ℵ−1M1G+ℱG.
Now, we give a lower bound for theℱ-coindex in terms of pendant vertices.Theorem 12.
SupposeG be a connected graph of order ℵ and r pendant vertices. Then,(25)ℱ¯G≥6ℵr−4r2−7r.Proof.
It is easy to see that our result holds forr=0. Now, we assume that r≥1. Here, we let that G has exactly one pendant vertex, called x, and y is the unique neighbor of x. Hence,(26)ℱ¯G≥∑z∈VG∖y,xζG2z+1≥∑w∈VG∖y,x5,=5ℵ−2.
Here, we can let thatr≥2. Each pair of pendant vertices contributes 2 to ℱ¯G. The total contribution of pendant vertex pairs to ℱ¯G is 2r2. Assume that x is a pendant vertex in G and y is its unique neighbor. Then, for any nonpendant vertex z such that w∈VG∖y,x, the contribution of vertex pairs x,z to ℱ¯G is 1+ζG2z. The total contribution of such vertex pairs x,z to ℱ¯G is ℵ−r−1r11+ζG2z. Note that ζGz≥2 for any nonpendant vertex z in G; therefore, we get ℱ¯G≥r21+1+ℵ−r−1r21+5=6ℵr−4r2−7r, as desired. □Theorem 13.
IfG is a t-regular graph of order ℵ, then(27)ℱG¯=ℵ−12ℵℵ−1−3tℵ−ℵt3+3ℵℵ−1t2,ℱ¯G¯=ℵt2ℵ−t−1,ℱ¯G¯=ℵtℵ−12−2tℵ−1+t2.Proof.
We know that anyt-regular graph of order ℵ has ℵt/2 edges.(1)
By applying Theorem10, we have(28)ℱG¯=2ℵ−12ϵ¯−2ϵ+3ℵ−1M1G−ℱG,=2ℵ−12ℵℵ−12−3ℵt2+3ℵ−1nt2−ℵt3,=ℵ−12ℵℵ−1−3tℵ−ℵt3+3ℵℵ−1t2.(2)
By applying Theorem9, we can write(29)ℱG¯=ℵ−1M1G−ℱG=ℵℵ−1t2−ℵt3.(3)
Similarly, by applying Theorem11, we obtain(30)ℱ¯G¯=2ℵ−12ϵ−2ℵ−1M1G+ℱG,=2ℵ−12ℵt2−2ℵt2ℵ−1+ℵt3,=ℵtℵ−12−2tℵ−1+t2.
Now, we give lower and upper bounds for theℱ-coindex.Theorem 14.
SupposeG be a connected ℵ,ϵ-graph, maximum degree be Θ, and nonpendant minimum degree be π1 and with s leaves. Then,(31)ℱ¯G≤sℵ−s−1Θ+1Θ+Θπ1+π1Θℵ22+s22+3s2−ℵ2−ϵ+ss−1−ℵsℵ−22,ℱ¯G≥sℵ−s−1π1+1π1+π1Θ+Θπ1ℵ22+s22+3s2−ℵ2−ϵ+ss−1−ℵs.Proof.
It can be easily seen that the number of vertices pairsxi,xj is as follows:(32)X1=xi,xj|xixj∉EG,X1=ℵℵ−12−ϵ,X2=xi,xj|xixj∉EGandζGxi=ζGxj=1,X2=ss−12,X3=xi,xj|xixj∉EGζGxi=1orζGxj=1,X3=sℵ−s−1,X4=xi,xj|xixj∉EGζGxi>1,andζGxj>1,X4=ℵℵ−12−ϵ−ss−12−sℵ−s−1.
For any vertexxi, we have π1≤ζGxi≤Θ and π12≤ζGxiζGxj≤Θ2 for nonpendant vertices. Therefore, xixj∉EG.
We continue the proof with the following four cases.
Letxixj∉EG.Case 1.
IfζGxi>1andζGxj>1, we have(33)ζGxiζGxj+ζGxjζGxi≤Θπ1+π1Θ.Case 2.
IfζGxi>1andζGxj=1, hence, we get(34)ζGxiζGxj+ζGxjζGxi≤Θ+1Θ.
It is clear that, for eachxixj∉EG, ζGxi,ζGxj≤ℵ−2. Hence, by applying the definition of the ℱ-coindex and above facts, we can write(35)ℱ¯Gℵ−22=∑xixj∉EGζG2xj+ζG2xjn−22≤∑xixj∉EGζG2xj+ζG2xjζGxiζGxj,=∑xixj∉EGζGxiζGxj+ζGxjζGxi,=∑xixj∉EG,ζGxi=1,ζGxj=1ζGxiζGxj+ζGxjζGxi+∑xixj∉EG,ζGxi>1,ζGxj=1ζGxiζGxj+ζGxjζGxi+∑xixj∉EG,ζGxi>1,ζGxj>1ζGxiζGxj+ζGxjζGxi≤∑xixj∉EG,ζGxi=1,ζGxj=12+∑xixj∉EG,ζGxi>1,ζGxj=1Θ+1Θ+∑xixj∉EG,ζGxi>1,ζGxj>1Θπ1+π1Θ,=ss−1+Θ+1Θsℵ−s−1+Θπ1+π1Θℵℵ−12−ϵ−ss−12−sℵ−s−1,=sℵ−s−1Θ+1Θ+Θπ1+π1Θℵ22+s22+3s2−n2−ϵ+ss−1−ℵs,as desired.Case 3.
IfζGxi>1andζGxj>1, we get(36)ζGxiζGxj+ζGxjζGxi≥π1Θ+Θπ1.Case 4.
IfζGxi>1andζGxj=1, we get that(37)ζGxiζGxj+ζGxjζGxi≥π1+1π1.
Thus,(38)ℱ¯G=∑xixj∉EGζG2xi+ζG2xj≥∑xixj∉EGζG2xi+ζG2xjζGxiζGxj,=∑xixj∉EGζGxiζGxj+ζGxjζGxi,=∑xixj∉EG,ζGxi=1,ζGxj=1ζGxiζGxj+ζGxjζGxi+∑xixj∉EG,ζGxi>1,ζGxj=1ζGxiζGxj+ζGxjζGxi+∑xixj∉EG,ζGxi>1,ζGxj>1ζGxiζGxj+ζGxjζGxi≥∑xixj∉EG,ζGxi=1,ζGxj=12+∑xixj∉EG,ζGxi>1,ζGxj=1π1+1π1+∑xixj∉EG,ζGxi>1,ζGxj>1π1Θ+Θπ1,=ss−1+π1+1π1sℵ−s−1+π1Θ+Θπ1ℵℵ−12−ϵ−ss−12−sℵ−s−1,=sℵ−s−1π1+1π1+π1Θ+Θπ1ℵ22+s22+3s2−ℵ2−ϵ+ss−1−ℵs,and the proof is completed.
By settings=0 in Theorem 14, we can obtain the following results.Corollary 1.
SupposeG be a connected ℵ,ϵ-graph, maximum degree be Θ, and nonpendant minimum degree be π1. Then,(39)ℱ¯G≤Θπ1+π1Θℵ22−ℵ2−ϵℵ−22,ℱ¯G≥π1Θ+Θπ1ℵ22−ℵ2−ϵ.
Here, we give a lower bound for theℱ-coindex.Proposition 1.
SupposeG be a connected graph of order ℵ and size ϵ. Then,(40)ℱ¯G≥ℵ2−2ϵ−ℵ.Proof.
By applying the definition of theℱ-coindex, we have(41)ℱ¯G=∑xixj∉EGζG2xi+ζG2xj≥∑xixj∉EG2=2ℵℵ−12−ϵ=ℵ2−2ϵ−ℵ.
In the following result, we obtain the lower bound for theℱ-coindex.Theorem 15.
SupposeG be a connected ℵ,ϵ-graph and ℵ≥2. Then,(42)ℱ¯G≥M¯15G+2M¯22G2ℵ−22.Proof.
Clearly, for eachxixj∉EG, we have maxζGxi,ζGxj≤ℵ−2. It follows that(43)2n−22ℱ¯G=2ℵ−22∑xixj∉EGζG2xi+ζG2xj≥∑xixj∉EGζG2xi+ζG2xj2,=∑xixj∉EGζG4xi+ζG4xj+2ζG2viζG2vj,=∑xixj∉EGζG4xi+ζG4xj+2∑xixj∉EGζGxiζGxj2,=M¯15G+2M¯22G,and the proof is completed.
Now, we give an upper bound for theℱ-coindex in terms of the second Zagreb coindex and the redefined first Zagreb coindex.Theorem 16.
SupposeG be a graph of order ℵ. Then,(44)ℱ¯G≤2ℵ−23ReZG¯1G−2ℵ−22M¯2G.Proof.
By applying the definition of theℱ-coindex, we have(45)ℱ¯Gn−22=∑xixj∉EGζG2xi+ζG2vjn−22≤∑vivj∉EGζGxiζGxj+ζGxjζGxi,=∑xixj∉EG1ζGxi+1ζGxjζGxj+ζGxj−2ζGxiζGxj,=∑xixj∉EG1ζGxi+1ζGxjζGxj+ζGxj−2∑xixj∉EGζGxiζGxj≤2n−2∑xixj∉EG1ζGxi+1ζGxj−2∑xixj∉EGζGxiζGxj,=2n−2∑xixj∉EGζGxi+ζGxjζGxiζGxj−2M¯2G,=2ℵ−2ReZG¯1G−2M¯2G,which leads to the result.
In the following results, we obtain upper bounds for theℱ-coindex in terms of the first and second Zagreb indices and hyper-Zagreb coindex.Theorem 17.
SupposeG be a ℵ,ϵ-graph. Then,(46)ℱ¯G+M¯2G≤3ϵ¯ℵ−22,(47)ℱ¯G+M¯1G≤2ϵ¯ℵ2−3ℵ+2,(48)ℱ¯G+HM¯G≤6ϵ¯ℵ−22.The equalities hold if and only ifG≅Kℵ or G is ℵ−2-regular or ℵ−2,ℵ−1-semiregular.Proof.
By applying the definitions of theℱ-coindex and second Zagreb coindex, we can write(49)ℱ¯G+M¯2G=∑xixj∉EGζG2xi+ζG2xj+ζGxiζGxj≤∑xixj∉EGℵ−22+ℵ−22+ℵ−22,=3ϵ¯ℵ−22.
Likewise, we have(50)ℱ¯G+M¯1G=∑xixj∉EGζG2xi+ζG2xj+ζGxi+ζGxj≤∑xixj∉EGℵ−22+ℵ−22+2ℵ−2,=2ϵ¯ℵ2−3ℵ+2,and(51)ℱ¯G+HM¯G=∑xixj∉EGζG2xi+ζG2xj+ζGxi+ζGxj2,=∑xixj∉EGζG2xi+ζG2xj+ζG2xi+ζG2xj+2ζGxiζGxj,=∑xixj∉EG2ζG2xi+2ζG2xj+2ζGxiζGxj≤∑xixj∉EG4ℵ−22+2ℵ−22,=6ϵ¯ℵ−22.
The equalities hold in (46)–(48) if and only if ζGxi=ζGxj=ℵ−2, for each xixj∉EG. This implies that each vertex of G has degree ℵ−1 or ℵ−2; that is, G≅Kℵ or G is ℵ−2-regular or ℵ−2,ℵ−1-semiregular.
By using Theorems1, 8, 10, and 9, we get the following result.Theorem 18.
SupposeG be a ℵ,ϵ-graph. Then,(52)ℱG¯≥2ℵ−12ϵ¯−2ϵ+3ℵ−14ϵ2ℵ+12Θ−δ2−2M2G+ϵℵ−2,ℱ¯G≥ℵ−14ϵ2ℵ+12Θ−δ2−2M2G+ϵℵ−2.The following results were obtained by combining Theorems1, 4, 5, 6, and 11.Theorem 19.
SupposeG be a ℵ,ϵ-graph. Then,(53)ℱ¯G¯≥2ϵℵ−12−2ℵ−14ϵ2ℵ+12Θ−δ2+16ϵ3ℵ2−2M2G,ℱ¯G¯≥2ϵℵ−12−2ℵ−14ϵ2ℵ+12Θ−δ2+M1G22ϵ,ℱ¯G¯≥2ϵℵ−12−2ℵ−14ϵ2ℵ+12Θ−δ2+Θ3+Θ23+Φ.The following results were obtained by combining Theorems2, 4, 10, and 9.Theorem 20.
SupposeG be a ℵ,ϵ-graph. Then,(54)ℱG¯≤2ℵ−12ϵ¯−2ϵ+3ℵ−14ϵ2ℵ+ℵ4Θ−δ2−16ϵ3ℵ2−2M2G,ℱ¯G≤ℵ−14ϵ2ℵ+ℵ4Θ−δ2−16ϵ3ℵ2−2M2G.
By applying Theorems2, 8, and 11, we can obtain the following theorem.Theorem 21.
SupposeG be a ℵ,ϵ-graph. Then,(55)ℱ¯G¯≤2ϵℵ−12−2ℵ−14ϵ2ℵ+ℵ4Θ−δ2+2M2G+ϵℵ−2.The following results were obtained by Theorems2, 5, 10, and 9.Theorem 22.
SupposeG be a ℵ,ϵ-graph. Then,(56)ℱG¯≤2ℵ−12ϵ¯−2ϵ+3ℵ−14ϵ2ℵ+ℵ4Θ−δ2−M1G22ϵ,ℱ¯G≤ℵ−14ϵ2ℵ+ℵ4Θ−δ2−M1G22ϵ.The following results were obtained by combining Theorems7, 10, and 9.Theorem 23.
SupposeG be a ℵ,ϵ-graph. Then,(57)ℱG¯≤2ℵ−12ϵ¯−2ϵ+3ℵ−14ϵ2ℵ+ℵ4Θ−δ2−Θ3+Θ23+Y,ℱ¯G≤ℵ−14ϵ2ℵ+ℵ4Θ−δ2−Θ3+Θ23+Y.


## 4. Conclusion

In this paper, we investigate the relationship between theℱ-coindex and the other topological coindices, such as the first and second Zagreb coindices, the hyper-Zagreb coindex, and the redefined first Zagreb coindex. However, there are still open and challenging problems for researchers, for example, the problem on the relationship among the ℱ-coindex and GA-coindex, harmonic coindex, Randić coindex, etc.



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