# Second Zagreb and Sigma Indices of Semi and Total Transformations of Graphs

Authors: Zhen Yang; Micheal Arockiaraj; Savari Prabhu; M. Arulperumjothi; Jia-Bao Liu
Journal: Complexity
 (2021)

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## Abstract

The study of structure-property relations including the transformations of molecules is of utmost importance in correlations with corresponding physicochemical properties. The graph topological indices have been used effectively for such study and, in particular, bond-based indices play a vital role. The bond-additive topological indices of a molecular graph are defined as a sum of edge measures over all edges in which edge measures can be computed based on degrees, closeness, peripherality, and irregularity. In this study, we provide the mathematical characterization of the transformation of a structure that can be accomplished by the novel edge adjacency and incidence relations. We derive the exact expressions of bond type indices such as second Zagreb, sigma indices, and their coindices of total transformation and two types of semitransformations of the molecules which in turn can be used to characterize the topochemical and topostructural properties.


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## Body

## 1. Introduction

Topological indices are graph invariants that play an important role in chemical and pharmaceutical sciences, since they can be used to predict physicochemical properties of organic compounds in view of successful applications in QSAR and QSPR techniques [1–5]. These indices are mainly classified into distance-based and degree-based. Development of such topological indices is of immense value in quantitative structure-activity relations. The first and second Zagreb indices were the oldest degree-based indices and found significant applications [6, 7]. The Zagreb indices have first appeared in the topological formula for the total π-energy of conjugated molecules and also useful in the study of anti-inflammatory activities of chemical instances. The generalization of the first Zagreb index is named as general sum-connectivity index [8] and there are many types of generalization and reformulation on the Zagreb indices based on vertex and edge degrees [8–11], in particular, the forgotten index is recently revisited with important applications to drug molecular structures [12, 13].It was known that most of the molecular structures are not regular and, hence, the quantitative measure based on irregularity is of great importance in mathematical chemistry. In the case of octane isomers, the application of various degree-based irregularity measures for the prediction of physicochemical properties such as boiling point, standard enthalpy of vaporization, acentric factor, enthalpy of vaporization, and entropy was tested and predicted with good accuracy [14]. As a result of which many topological indices of this kind have been discussed and a few of them are Collatz–Sinogowitz, degree variance, discrepancy, Albertson, Bell, and total irregularity and sigma indices [14–17].The Albertson index is the most commonly used irregularity measures that provide the structural perfection of chemical compounds. For this purpose, the imbalance of an edge is defined as the absolute difference between the degrees of end vertices and the summation is taken over all edges. In this paper, we focus our attention on the recently popular, sigma index, which is defined as the sum of squares of imbalance of every edge. Moreover, there is a nice relationship between second Zagreb, forgotten, and sigma indices which states that the difference between forgotten and sigma indices is twice the second Zagreb index [18] and some properties of the sigma index discussed in [19].The structure of a molecular graphG can be transformed into another graph TG by imposing desired rules based on the original structure of G so that there is a one-to-one correspondence between original graph G and the transformation graph TG. Such a transformation of graphs and their characterization was attempted by many researchers in chemical graph theory [20–26] because the complex structure of transformation graph can be easily analyzed by the original graph. For instance, the first Zagreb index [21, 25], second Zagreb index [21, 27], forgotten index [20, 28] of transformation graphs, and Zagreb indices of transformation of line graph of subdivision graphs [29] were discussed. In this, we observe that the entire process of the second Zagreb index [27] was wrongly dealt and we will discuss with details in Section 3. Moreover, the forgotten index [20, 28] of transformation of graphs was considered with vertex a-Zagreb and (a, b)-Zagreb indices. In this study, we give the correct expressions for the second Zagreb index of transformation graphs and rewrite for the forgotten index via general sum-connectivity index. Finally, we derive the analytical expressions for the sigma index of two types of semitransformations and a total transformation.Throughout this paper, we writeG to denote a simple connected graph with vertex set VG and edge set EG. The number of elements in the vertex set and the edge set, respectively, is denoted by n (order) and m (size). The number of edges incident with a vertex s∈VG is called the degree of the vertex s, denoted by dGs. The neighborhood of a vertex s, denoted by NGs, is a set of all vertices which are adjacent to s. Two edges e,f∈EG are said to be adjacent if they share a common vertex and we write as e∼f and in case they are not adjacent, e≁f. In the same line of notation, s∈VG, f∈EG, and s∼f mean that s is an end vertex of f while s≁f that s is not an end vertex of f. The degree of an edge e=st, denoted by dGe, is the number of edges that are adjacent to e, i.e., dGe=dGs+dGt−2. The complement of a graph G, represented by G¯, is a graph obtained from G with the same vertex set of G such that s is adjacent to t in G¯ if and only if s is not adjacent to t in G. Hence, the size of G¯ is 1/2n2−n−2m, and the degree of each vertex s∈VG¯ is dG¯s=n−dGs−1.We close this section by listing down (in Table1) certain bond-additive topological indices [7–13, 18, 28, 30, 31] and their coindices which are needed for our study.Table 1
Bond-additive indices ofG.
ItemIndexCoindexFirst ZagrebM1G=∑st∈EGdGs+dGtM1¯G=∑st∉EGdGs+dGtSecond ZagrebM2G=∑st∈EGdGsdGtM2¯G=∑st∉EGdGsdGtForgottenFG=∑st∈EGdGs2+dGt2F¯G=∑st∉EGdGs2+dGt2Sum-connectivityχαG=∑st∈EGdGs+dGtαχα¯G=∑st∉EGdGs+dGtαReformulated first ZagrebEM1G=∑e,f∈EGe∼fdGe+dGfEM1¯G=∑e,f∈EGe≁fdGe+dGfReformulated second ZagrebEM2G=∑e,f∈EGe∼fdGedGfEM2¯G=∑e,f∈EGe≁fdGedGfSigmaσG=∑st∈EGdGs−dGt2σ¯G=∑st∉EGdGs−dGt2


## 2. Transformation Graphs

The concept of transformation graphs is to construct a new graph from the original graphG based on the structural connectivity. Generally, we can transform the original graph by imposing any combinations of the following:Forα,β,γ∈+,−, vi∈VG, 1≤i≤n, and ej∈EG, 1≤j≤m,(1)
vi,vj∈VG, vi is adjacent to vj in G if α=+ and vi is not adjacent to vj in G if α=−(2)
ei,ej∈EG, ei is adjacent to ej in G if β=+ and ei is not adjacent to ej in G if β=−(3)
vi∈VG and ej∈EG, ej is incident to vi in G if γ=+ and ej is not incident to vi in G if γ=−The type-I semitransformation of a graphG, denoted by T1αγG, is a graph with the vertex set VG∪EG, and for s,t∈VT1αγG, s and t are adjacent in T1αγG if and only if (#1) and (#3) hold [21]. Following this, it is natural to define another semitransformation, called type-II semitransformation and denoted by T2βγG, whose vertex set is VG∪EG, and for s,t∈VT2βγG, s and t are adjacent in T2βγG if and only if (#2) and (#3) hold. The total transformation graph TαβγG is a graph with the same vertex set as above VG∪EG, and for s,t∈VTαβγG, s and t are adjacent in TαβγG if and only if (#1), (#2), and (#3) hold [32].The concept of semitotal point, semitotal line, and total graphs came into the literature earlier [33, 34] and these three graphs are particular cases of our T1αγG, T2βγG, and TαβγG, i.e., T1++G is the semitotal point graph, T2++G is the semitotal line graph, and T+++G is the total graph. Since there are four distinct 2-permutations of +,−, we can construct totally eight different graphs from two types of semitransformations. For a graph G depicted in Figure 1, the two types of semitransformation graphs are shown in Figure 2. In the same way, there are eight distinct 3-permutations of +,− and again totally eight graphs can be constructed from the total transformation in which T−−−G≅T+++G¯, T−−+G≅T++−G¯, T−+−G≅T+−+G¯, and T−++G≅T+−−G¯. For the same graph in Figure 1, the eight classes of total transformation graphs are given in Figure 3.Figure 1
The graphG.Figure 2
(a)T1++G; (b) T1+−G; (c) T1−+G; (d) T1−−G; (e) T2++G; (f) T2+−G; (g) T2−+G; (h) T2−−G.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 3
(a)T+++G; (b) T++−G; (c) T−+−G; (d) T−++G; (e) T−−−G; (f) T−−+G; (g) T+−+G; (h) T+−−G.
(a)(b)(c)(d)(e)(f)(g)(h)Lemma 1 (see [21]).
LetG be graph with n and m as its order and size, respectively. Then, the order of T1αγG is m+n, and the size is(1)ET1αγG=3m,:α=+,γ=+,mn−1,:α=+,γ=−,12nn−1+m,:α=−,γ=+,12nn−1+mn−3,:α=−,γ=−.Lemma 2.
LetG be graph with n and m as its order and size, respectively. Then, the order of T2βγG is m+n, and the size is(2)ET2βγG=12M1G+m,:β=+,γ=+,12M1G+mn−3m,:β=+,γ=−,12m2+5m−M1G,:β=−,γ=+,12m2+2mn−3m−M1G,:β=−,γ=−.Lemma 3.
LetG be graph with n and m as its order and size, respectively. Then, the order of TαβγG is m+n, and the size is(3)ETαβγG=12M1G+4m,:α=+,β=+,γ=+,12M1G+2mn−2,:α=+,β=+,γ=−,12m2+7m−M1G,:α=+,β=−,γ=+,12M1G+nn−1,:α=−,β=+,γ=+,12m+n2−5m−n−M1G,:α=−,β=−,γ=−,12m2+nn−1+3m−M1G,:α=−,β=−,γ=+,12M1G+2mn−4+nn−1,:α=−,β=+,γ=−,12m2+2mn−m−M1G,:α=+,β=−,γ=−.We now recall the results pertaining to the first and second Zagreb indices of type-I semitransformation graphs and the first Zagreb index of total transformation graph which are helpful for our study.Lemma 4 (see [21, 25]).
LetG be a graph with order n and size m. Then,(i)
M1T1++G=4m+M1G(ii)
M1T1+−G=nm2+mn−22(iii)
M1T1−+G=nn−12+4m(iv)
M1T1−−G=4M1G+mn−22+m+n−1nm+n−1−8mLemma 5 (see [21]).
LetG be a graph with order n and size m. Then,(i)
M2T1++G=4M1G+4M2G(ii)
M2T1+−G=m3+m2n−22(iii)
M2T1−+G=1/2nn−13−2mn−12+8mn−1(iv)
M2T1−−G=1/24n−2M1G−4m+n−1M1¯G+8M2¯G+m+n−12n2−n−2m+2mm+n−1n−22−8m2n−2Lemma 6 (see [25]).
LetG be a graph with order n and size m. Then,(i)
M1T+++G=4M1G+2M2G+FG(ii)
M1T++−G=mnm+n−8+16m+2n−4M1G+2M2G+FG(iii)
M1T+−+G=mm+32−2m+1M1G+2M2G+FG(iv)
M1T−++G=nn−12+2M2G+FG(v)
M1T−−−G=m+nm+n2−10m−2n+1+8m−2m+n−3M1G+2M2G+FG(vi)
M1T−−+G=nn−12+mm+32−2m+6M1G+2M2G+FG(vii)
M1T−+−G=mm+32+m+nm+n−12−2m2+7mm+n−1+2n−2M1G+2M2G+FG(viii)
M1T+−−G=mmn+1+m+nm+n−2−2m+n−1M1G+2M2G+FG


## 3. Main Results

In this section, we derive the analytic expressions for the sigma index and coindex of semi and total transformations of graphs. Bearing the relationσG=FG−2M2G in mind, we first study the second Zagreb index and then the forgotten index and finally deduce the results for the sigma index.


### 3.1. Second Zagreb Index of Transformation Graphs

The second Zagreb index of total transformation of graphs was expressed in [27], and by careful inspection, we notice that the entire process is vague and results in incorrect expressions. For instance, it was proved [27] that M2T+++G=8M1G+6M2G+FG. Suppose G=Pn, a path on n vertices. Then, T+++Pn is a graph on 2n−1 vertices and 4n−5 edges in which 2 vertices of degrees 2 and 3 each and 2n−5 vertices of degree 4 while 2 edges with degrees of end vertices 2,3 and 2,4each, and 4 edges with 3,4 and 4n−13 edges with 4,4. Hence, M2T+++G=6×2+8×2+12×4+16×4n−13=64n−132. However, M1Pn=4n−6, M2Pn=4n−8, and FPn=8n−14, resulting that 8M1Pn+6M2Pn+FG=64n−110. Hence, we now compute the correct analytic expressions of the second Zagreb index and coindex of total transformation graphs using reformulated Zagreb indices. Moreover, the type-II semitransformation is newly introduced in this paper, and hence we also obtain the exact expressions for first and second Zagreb indices. The following theorem gives the exact expression for second Zagreb indices of first four transformations in terms of edge version of first and second Zagreb indices of the arbitrary graph.Theorem 1.
LetG be a graph with order n and size m. Then,(i)
M2T+++G=EM2G+2EM1G+8M2G+2M1G+2FG−4m(ii)
M2T++−G=EM2G+n−2EM1G+1/2n−22+2mn−2M1G+m3+m2n−2n−4−mn−22(iii)
M2T+−+G=EM2¯G−m+1EM1¯G+1/2mm+13−1/2m2−2m−11M1G−2FG(iv)
M2T−++G=EM2G+2EM1G+2nM1G+1/2nn−13−mn2−2n+5Proof.
The graphT+++G has m+n vertices and 1/2M1G+2m edges in which m edges are actual edges in G by condition (#1), 1/2M1G−m edges are produced by condition (#2) called edge adjacency relation edges (line graph edges), and 2m edges are edges produced by condition (#3) called incidence relation edges. For any vertex s∈VT+++G,(4)dT+++Gs=2dGs,ifs∈VG,dGs+2,ifs∈EG.
Therefore,(5)M2T+++G=∑st∈ET+++GdT+++GsdT+++Gt=∑st∈ET+++G∩EGdT+++GsdT+++Gt+∑st∈ET+++G∩ELGdT+++GsdT+++Gt+∑st∈ET+++G∖EG∪ELGdT+++GsdT+++Gt=∑st∈EG2dGs2dGt+∑s,t∈EGs∼tdGs+2dGt+2+∑s∈VG,t∈EGs∼t2dGsdGt+2=4M2G+∑s,t∈EGs∼tdGsdGt+∑s,t∈EGs∼t2dGs+dGt+4ELG+∑s∈VG∑x∈NGs2dGsdGs+dGx=4M2G+EM2G+2EM1G+4ELG+2∑s∈VGdGs3+2∑s∈VG∑x∈NGsdGsdGx=4M2G+EM2G+2EM1G+4M1G2−m+2FG+4M2G=EM2G+2EM1G+8M2G+2M1G+2FG−4m.
This completes the proof of assertion (i). Next, for any vertexs∈VT++−G,(6)dT++−Gs=m,ifs∈VG,dGs+n−2,ifs∈EG.
It can be seen that(7)M2T++−G=∑st∈EGdGsdGt+∑s,t∈EGs∼tdGsdGt+∑s∈VG,t∈EGs≁tdGsdGt=∑st∈EGm⋅m+∑s,t∈EGs∼tdGs+n−2dGt+n−2+∑s∈VG,t∈EGs≁tmdGt+n−2=m3+EM2G+n−2EM1G+n−22ELG+∑t∈EG∑s∈VGs≁tmdGt+n−2=m3+EM2G+n−2EM1G+n−22M1G2−m+m∑t∈EGn−2dGt+n−2=m3+EM2G+n−2EM1G+12n−22M1G−2mn−22+mn−2∑t∈EGdGt+m2n−22=m3+EM2G+n−2EM1G+12n−22M1G−2mn−22+mn−2M1G−2m+m2n−22=EM2G+n−2EM1G+12n−22+2mn−2M1G+m3+m2n−2n−4−mn−22.
To complete the proof of assertion (iii), we notice that for any vertex,s∈VT+−+G,(8)dT+−+Gs=2dGs,ifs∈VG,m+1−dGs,ifs∈EG.
As before, we can easily write that(9)M2T+−+G=∑st∈EG2dGs2dGt+∑s,t∈EGs≁tm+1−dGsm+1−dGt+∑s∈VG,t∈EGs∼t2dGsm+1−dGt=4M2G+m+12mm−12−ELG−m+1EM1¯G+EM2¯G+2∑s∈VG∑x∈NGsdGsm+3−dGs+dGx=EM2¯G−m+1EM1¯G+4M2G+12mm+13−m+12M1G+2m+3M1G−2FG−4M2G=EM2¯G−m+1EM1¯G+12mm+13−12m2−2m−11M1G−2FG.
The final assertion follows from the fact that for any vertexs∈VT−++G,(10)dT−++Gs=n−1,ifs∈VG,dGs+2,ifs∈EG.Theorem 2.
LetG be a graph with order n and size m. Then,(1)
M2T−−−G=1/2M12G−3m+n−12−82m+n−2M1G+4m+4n−22M2G+2m+2n−7FG−2EM2G−4EM1G+m+nm+n−13−12mm+n−12+8m2m+1(2)
M2T−−+G=1/2M12G−3m2+14m+12n−17M1G+4m+4n−6M2G+2m+2n−3FG−2EM2G−2n−2EM1G+m4+7m3+4m2n+11m2−2mn2+24mn−29m+n4−3n3+3n2−n(3)
M2T−+−G=1/2M12G+3m+n−12−2m2+7m−2m+12m+2n−3+m2−2m−11M1G+22m+2n−3M2G+2m+2n+1FG−2EM2¯G+2m+1EM1¯G+m+nm+n−13−3m+n−12m2+7m+m2+7m2+mm+322m+2n−3−mm+13(4)
M2T+−−G=1/2M12G+2nn−1−3m+n−12−4nM1G+22m+2n−3M2G+2m+2n−3FG−2EM2G−4EM1G+m+nm+n−13−3nn−1m+n−12+n2n−12+nn−122m+2n−3−nn−13+2mn2−2n+5Proof.
It was proved [35] that(11)M2G¯=12nn−13−3mn−12+2m2+2n−32M1G−M2G,and known that T+++G¯≅T−−−G, T++−G¯≅T−−+G, T+−+G¯≅T−+−G, and T+−−G¯≅T−++G. By Lemma 6 and Theorem 1, we can easily complete the proof.The Zagreb coindices are introduced in [36] with extensive applications in the field of chemical graph theory and widely discussed in [9, 10, 37–39]. Therefore, it will be worth finding the second Zagreb coindices of total transformations.Theorem 3.
LetG be a graph with order n and size m. Then,(1)
M2¯T+++G=1/2M12G+8m−1M1G−18M2G−5FG−2EM2G−4EM1G+8m2m+1(2)
M2¯T++−G=1/2M12G+2mn−4m−n2+2n+4M1G−2M2G−FG−2EM2G−2n−2EM1G+2m2n2−2m3−5m2n+mn2−8m(3)
M2¯T+−+G=1/2M12G−m2+14m+9M1G−2M2G+3FG−2EM2¯G+2m+1EM1¯G+10m3+40m2−10m(4)
M2¯T−++G=1/2M12G+2nn−3M1G−2M2G−FG−2EM2G−4EM1G+2mn2−2n+5(5)
M2¯T−−−G=1/22EM2G+4EM1G+m2+n2+2mn−10m−10n+13M1G−4m+n−5M2G−2m+n−3FG+4m3+8m2n−8m2+4mn2−8mn−4m(6)
M2¯T−−+G=1/22EM2G+2n−2EM1G+m2−2n2+10m+14n−11M1G−4m+n−1M2G−2m+n−1FG−2m3+2m2n2−6m2n−8m2+8mn2−30mn+20m(7)
M2¯T−+−G=1/22EM2¯G−2m+1EM1¯G+2m2−n2+2mn+4m+6n+6M1G−4m+n−1M2G−2m+n+1FG+m2n2+7mn2−32mn−2m3−16m2+26m(8)
M2¯T+−−G=1/22EM2G+4EM1G+m2+2mn−2m+n2+2n+1M1G−4m+n−1M2G−2m+n−1FG+m2n2−2mn2−m2n+4mn−10mProof.
It was shown in [35] that(12)M2¯G=2m2−12M1G−M2G,and combining the results of Lemma 6 and Theorem 1, we can finish the proof by simple mathematical calculations.The following theorem fills the gap in the literature with respect to the results found in [21, 25].Theorem 4.
LetG be a graph with order n and size m. Then,(1)
M1T2++G=M1G+FG+2M2G [25](2)
M1T2+−G=2M2G+FG+2n−7M1G+mnm+n−4mm+2n−4(3)
M1T2−+G=2M2G+FG−2m+5M1G+mm+32(4)
M1T2−−G=2M2G+FG−2m+2n−3M1G+m2n−4+mm+n−12(5)
M1¯T2++G=m+n−2M1G−2M2G−FG+2mn+2mm−1 [25](6)
M1¯T2+−G=m−n+6M1G−2M2G−FG+mnm+n−2mm+5(7)
M1¯T2−+G=m−n+6M1G−2M2G−FG+mnm+5−2mm+7(8)
M1¯T2−−G=m+n−2M1G−2M2G−FG+mnn−3+2mm+1Proof.
From the construction of type-II semitransformation, it is easily seen that for any vertexs∈VT2βγG such that s∈VG,(13)dT2βγGs=dGs,:β=+,γ=+,dGs,:β=−,γ=+,m−dGs,:β=+,γ=−,m−dGs,:β=−,γ=−.
In the same way, for any vertexs∈VT2βγG such that s∈EG,(14)dT2βγGs=dGs+2,:β=+,γ=+,m+1−dGs,:β=−,γ=+,n−2+dGs,:β=+,γ=−,m+n−3−dGs,:β=−,γ=−.
The proof follows from routine mathematical simplifications and, in addition, using the relationM1¯G=2mn−1−M1G [35].In [25], the authors have made an attempt to find the second Zagreb index of type-I semitransformation and left the calculations of type-II semitransformation due to its computational complexity. The following theorem gives the exact analytical expressions of the second Zagreb indices for type-II transformations of an arbitrary graph.Theorem 5.
LetG be a graph with order n and size m. Then,(1)
M2T2++G=EM2G+2EM1G+2M2G+2M1G+FG−4m(2)
M2T2+−G=EM2G+n−2EM1G+2M2G+1/2n2+2mn−8m−2n−4M1G+FG+m2n−42−mn−22(3)
M2T2−+G=EM2¯G−m+1EM1¯G−2M2G−1/2m2−5M1G−FG+1/2mm+13(4)
M2T2−−G=EM2¯G−m+n−3EM1¯G−1/2m+n−32+2mn−10m−2n+2M1G−2M2G−FG+1/2mm+1m+n−32+2m2n−4m+n−1(5)
M2¯T2++G=1/2M12G+4m−5M1G−3FG−2EM2G−4EM1G−6M2G+4mm+2(6)
M2¯T2+−G=1/2M12G+2mn−n2−4m+11M1G−3FG−2EM2G−2n−2EM1G−6M2G+mn2mn−9m+n+8mm−1(7)
M2¯T2−+G=1/2M12G−mm+8M1G+2M2G−2EM2¯G+2m+1EM1¯G+FG+2m3m2+8m−5(8)
M2¯T2−−G=1/2M12G−m2−n2+8m+6n−8M1G+2M2G+FG−2EM2¯G+2m+n−3EM1¯G+mnmn−m−2n+8+2m3m2+2m−5Proof.
The proof of (i)–(iv) is similar to Theorem1, and for the sake the completeness, we give the proof of (i). The graph T2++G has m+n vertices and 1/2M1G+m edges in which 1/2M1G−m edges are produced by condition (#2) called edge adjacency relation edges (line graph edges) and 2m edges are edges produced by condition (#3) called incidence relation edges. Also, for any vertex, s∈VT2++G,(15)dT2++Gs=dGs,ifs∈VG,dGs+2,ifs∈EG.
Hence,(16)M2T2++G=∑st∈ET2++GdT2++GsdT2++Gt=∑st∈ET2++G∩ELGdT2++GsdT2++Gt+∑st∈ET2++G∖ELGdT2++GsdT2++Gt=∑s,t∈EGs∼tdGs+2dGt+2+∑s∈VG,t∈EGs∼tdGsdGt+2=∑s,t∈EGs∼tdGsdGt+∑s,t∈EGs∼t2dGs+dGt+4ELG+∑s∈VG∑x∈NGsdGsdGs+dGx=∑s,t∈EGs∼tdGsdGt+∑s,t∈EGs∼t2dGs+dGt+4ELG+∑s∈VGdGs3+∑s∈VG∑x∈NGsdGsdGx=EM2G+2EM1G+4M1G2−m+FG+2M2G=EM2G+2EM1G+2M2G+2M1G+FG−4m.
To complete the remaining parts, we apply equation (12) with the help of Theorem 4.


### 3.2.F-Index of Transformation Graphs

The forgotten index and coindex of type-I semi and total transformations of graphs have been obtained [20, 28] in terms of first Zagreb, second Zagreb, vertex a-Zagreb, and (a, b)-Zagreb indices. In this section, we rewrite vertex a-Zagreb and (a, b)-Zagreb indices in terms of the sum-connectivity index. Before proceeding to this, we shall state a basic lemma.Lemma 7 (see [28]).
LetG be a connected graph of order n and size m. Then,(i)
FG¯=nn−13−FG−6mn−12+3n−1M1G(ii)
F¯G=n−1M1G−FGThe following theorem is crucial for finding the sigma index of the transformation of an arbitrary graph.Theorem 6.
LetG be a connected graph of order n and size m. Then,(1)
FT+++G=8FG+χ3G(2)
FT++−G=χ3G+3n−12FG+2M2G+3n−42M1G+mn−43+m3n(3)
FT+−+G=3m+17FG−χ3G+mm+33−3m+32M1G+6m+3M2G(4)
FT−++G=nn−13+χ3G(5)
FT−−−G=6m+n−1M2G−3m2−18m+6mn−18n+3n2+15M1G+3m+3n−11FG−χ3G+m+nm+n−13−12mm+n−12(6)
FT−−+G=3m+9FG−χ3G−3m2+18m+27M1G+6m+18M2G+m4+9m3+27m2+27m+n4−3n3+3n2−n(7)
FT−+−G=χ3G+3n−20FG+3n2−12n+12m+36M1G+6n−24M2G+m+nm+n−13−mm+33−3m+n−12m2+7m+3mm+32m+n−1(8)
FT+−−G=3m+n−1FG+6m+n−1M2G−3m+n−12M1G−χ3G+m+nm+n−13−nn−13−3mnn−1m+n−1Proof.
The proof of (i)–(iv) can be derived using the degrees of vertices from the proof of Theorem1 and the remaining parts from Lemma 7.The following theorem is an easy consequence of combining Lemma7 and Theorem 6, which will be used to compute the analytical expressions of the forgotten coindices of total transformations of an arbitrary graph.Theorem 7.
LetG be a connected graph of order n and size m. Then,(1)
F¯T+++G=4m+n−1M1G+2m+n−1M2G+m+n−9FG−χ3G(2)
F¯T++−G=m−2n+11FG−χ3G−n2−2mn+8m−14n+40M1G+2m−4n+22M2G+2m2n2−9m2n+16m2+3mn2−24mn+48m(3)
F¯T+−+G=n−2m−18FG+χ3G+m2+18m−2mn−2n+29M1G+2n−4m−20M2G+m3n−4m3+6m2n−24m2+9mn−36m(4)
F¯T−++G=m+n−1FG+2m+n−1M2G−χ3G+mn3−2mn2+mn(5)
F¯T−−−G=χ3G−2m+2n−10FG+m2+n2+2mn−10m−10n+9M1G−4m+n−1M2G+4m3+8m2n−8m2+4mn2−8mn+4m(6)
F¯T−−+G=χ3G+m2+14m−2mn−6n+33M1G−4m−2n+20M2G−2m−n+10FG+m3n−4m3+6m2n−24m2+mn3−2mn2+10mn−36(7)
F¯T−+−G=2m−2n+11M2G−χ3G−n2−2mn+16m−6n+32M1G+m−2n+19FG+4m3+m2n2+8m2+7mn2−32mn+52m(8)
F¯T+−−G=χ3G−4m+n−1M2G−2m+n−1FG+m+n−12M1G+m2n2−m2nTheorem 8 (see [20]).
LetG be a connected graph of order n and size m. Then,(1)
FT1++G=8FG+8m(2)
FT1+−G=nm3+mn−23(3)
FT1−+G=nn−13+8m(4)
FT1−−G=12m+n−1M1G−8FG+nm+n−13−12mm+n−12+mn−23(5)
F¯T1++G=4mm+n−1−8m+4m+n−1M1G−8FG(6)
F¯T1+−G=m+n−1nm2+mn−22−nm3−mn−23(7)
F¯T1−+G=m+n−14m+nn−12−nn−13−8m(8)
F¯T1−−G=8FG−8m+n−1M1G+mm+n−1n−22+m+n−12nm+n−1+4m−nm+n−13−mn−23Theorem 9.
LetG be a connected graph of order n and size m. Then,(1)
FT2++G=FG+χ3G(2)
FT2+−G=m3n−6+mn−43+χ3G+3n−13FG+6n−4M2G+3n−42+3mM1G(3)
FT2−+G=FG+mm+33−3m+32M1G+3m+3FG+2M2G−χ3G(4)
FT2−−G=m3n−6+mm+n−13+3m+3n−4FG+6m+n−1M2G+3m−3m+n−12M1G−χ3G(5)
F¯T2++G=m+n−1M1G+2M2G+m+n−2FG−χ3G(6)
F¯T2+−G=2m−4n+22M2G+2mn−n2+15n−10m−41M1G+m−2n+12FG−χ3G+m+n−1mnm+n−4mm+2n−4−m3n−6−mn−43(7)
F¯T2−+G=2n−4m−20M2G+n−2m−11FG+m2+15m−2mn−5n+32M1G+χ3G+mm+32n−4(8)
F¯T2−−G=χ3G−4m+n−1M2G+m2+2mn−4m−n+n2M1G−2m+2n−3FG+2m3+m2n2−5m2n+4m2Proof.
The proof of the theorem follows from using the degrees of vertices as given in the proof of Theorem4 and Lemma 7.


### 3.3.σ-Index of Transformation Graphs

In this section, we first derive a relation betweenσ-index of a graph and its coindex. Following this, we derive another relation between σ-coindex of a graph and σ-index of the complement graph. Finally, we list down the σ-index and coindex of semi and total transformations of graphs from the above subsections. The following theorem gives the relationship between the sigma index and its coindex.Theorem 10.
LetG be any graph with n vertices and m edges. Then,(17)σG+σ¯G=nM1G−4m2.Proof.
The proof is completed from the definitions ofσ-index and coindex as explained in the following:(18)σG+σ¯G=∑st∈EGdGs−dGt2+∑st∉EGdGs−dGt2=∑s,t⊆VGdGs−dGt2=∑s,t⊆VGdGs2+dGt2−2dGsdGt=∑s,t⊆VGdGs2+dGt2−∑s,t⊆VG2dGsdGt=FG+F¯G−2M2G−2M2¯G=n−1M1G−2M2G−22m2−M2G−M1G2=nM1G−4m2.Corollary 1.
LetG be any graph with n vertices and m edges. Then,(19)σ¯G=nM1G+2M2G−FG−4m2.The following theorem establishes interesting result that the sigma index of the complement of a graph and sigma coindex of a graph is one and the same.Theorem 11.
LetG be any graph with n vertices and m edges. Then,(20)σG¯=σ¯G.Proof.
For any vertexs∈VG, dG¯s=n−1−dGs, and we have(21)σG¯=∑st∈EG¯dG¯s−dG¯t2=∑st∉EGn−1−dGs−n−1−dGt2=∑st∉EGdGt−dGs2=σ¯G.Corollary 2.
LetG be any graph with n vertices and m edges. Then,(22)σ¯G¯=σG.The main objective of this section is the following theorem.Theorem 12.
LetG be a connected graph of order n and size m. Then,(1)
σT+++G=4FG+χ3G−2EM2G−4EM1G−16M2G−4M1G+8m(2)
σT++−G=χ3G+3n−12FG+2M2G−n−22+2mn−2−3n−42M1G−2EM2G−2n−2EM1G−2m3−2m2n−2n−4+2mn−22+mn−43+m3n(3)
σT+−+G=12FG−χ3G+3m+3χ2G−2EM2¯G+2m+1EM1¯G+m2−2m−11−3m+32M1G+mm+33−mm+13(4)
σT−++G=χ3G−2EM2G−4EM1G−4nM1G+2mn2−2n+5(5)
σT−−−G=2EM2G+4EM1G−M12G+4n−m+1M1G+2m+n+8M2G+m+n−4FG−χ3G−8m2m+1(6)
σT−−+G=2EM2G+2n−2EM1G−M12G−χ3G+12n−4m−44M1G+2m−4n+24M2G+m−2n+12FG+2m3+16m2−4m2n+2mn2−24mn+56m(7)
σT−+−G=2EM2¯G−2m+1EM1¯G−M12G+χ3G+2m2+32m−2mn+38−2nM1G+2n−4m−18M2G+n−2m−21FG+m3n−14m3+6m2n−64m2+9mn−26m(8)
σT+−−G=2EM2G+4EM1G−M12G−χ3G−2nn−3M1G+2m+2nM2G+m+nFG+n3m−4n2m+5nm−10mIn sequence to Theorems3 and 7, we have the following.Theorem 13.
LetG be a connected graph of order n and size m. Then,(1)
σ¯T+++G=2EM2G+4EM1G+2m+n+8M2G+4n−m+1M1G+m+n−4FG−χ3G−M12G−16m2−8m(2)
σ¯T++−G=2EM2G+2n−2EM1G+2m−2n+12M2G+43n−m−11M1G+m−2n+12FG−χ3G−M12G+2m3−4m2n+16m2+2mn2−24mn+56m(3)
σ¯T+−+G=2EM2¯G−2m+1EM1¯G+χ3G−M12G+2n−4m−18M2G+2m2+32m−2mn+38−2nM1G+n−2m−21FG+m3n−14m3+6m2n−64m2+9mn−26m(4)
σ¯T−++G=2EM2G+4EM1G+2m+nM2G−2nn−3M1G+m+nFG−χ3G−M12G+mnn−12−2mn2−2n+5(5)
σ¯T−−−G=σT+++G(6)
σ¯T−−+G=σT++−G(7)
σ¯T−+−G=σT+−+G(8)
σ¯T+−−G=σT−++GThe following theorems give the exact expressions of the sigma index of type-I and type-II semitransformations.Theorem 14.
LetG be graph with n and m as its order and size, respectively. Then,(1)
σT1++G=8FG−8M1G−8M2G+8m(2)
σT1+−G=nm3+mn−23−2m3−2m2n−22(3)
σT1−+G=2mn−12−8mn−1+8m(4)
σT1−−G=4m+n−1M1¯G−8FG+43m+2n−1M1G−8M2¯G+m3n−8mn2−6m2n+17mn−10m3−4m2−10m(5)
σ¯T1++G=4m+n+2M1G+8M2G−8FG−4m8m−n+2(6)
σ¯T1+−G=2m2m−2n+4+2mnn−4+8m(7)
σ¯T1−+G=mn3−8mn2+21mn−18m(8)
σ¯T1−−G=8FG−42m+n−1M1G−4m+n−1M1¯G+8M2¯G+2m2m+4n−10+2m4n2−8n+5Theorem 15.
LetG be graph with n and m as its order and size, respectively. Then,(1)
σT2++G=χ3G−2EM2G−4EM1G−4M2G−4M1G−FG+8m(2)
σT2+−G=χ3G+3n−5FG+23n−14M2G+2n2−2mn+11m−22n+52M1G−2EM2G−2n−2EM1G+m3n+mn3−6m3−2m2n2+16m2n−10mn2−32m2+40mn−56m(3)
σT2−+G=2m+1EM1¯G−2EM2¯G−2m2+18m+32M1G+3m+4FG+23m+11M2G−χ3G+6m3+24m2+26m(4)
σT2−−G=2m+n−3EM1¯G−2EM2¯G−2m2+2mn+2n2+7m+2n−8M1G+3m+3n−2FG+6m+6n−2M2G−χ3G+4m3+mn3−4mn2+8m2n+9mn−8m2−10m(5)
σ¯T2++G=2EM2G+4EM1G−3m−n−4M1G−M12G+m+n+1FG+2m+n+2M2G−χ3G−4m2−8m(6)
σ¯T2+−G=2EM2G+2n−2EM1G−M12G−χ3G+15n−6m−52M1G+2m−4n+28M2G+m−2n+15FG+2m3−4m2n+12m2+2mn2−24mn+56m(7)
σ¯T2−+G=2EM2¯G−2m+1EM1¯G−M12G+χ3G+2m2−2mn+23m−5n+32M1G−22m−n+11M2G−2m−n+12FG+m3n−10m3+6m2n−40m2+9mn−26m(8)
σ¯T2−−G=2EM2¯G−2m+n−3EM1¯G+χ3G−M12G+2m2+2mn+4m+5n−8M1G−22m+2n−1M2G−2m+n−1FG−4m3−4m2n+2mn2−8mn+10m


## 3.1. Second Zagreb Index of Transformation Graphs

The second Zagreb index of total transformation of graphs was expressed in [27], and by careful inspection, we notice that the entire process is vague and results in incorrect expressions. For instance, it was proved [27] that M2T+++G=8M1G+6M2G+FG. Suppose G=Pn, a path on n vertices. Then, T+++Pn is a graph on 2n−1 vertices and 4n−5 edges in which 2 vertices of degrees 2 and 3 each and 2n−5 vertices of degree 4 while 2 edges with degrees of end vertices 2,3 and 2,4each, and 4 edges with 3,4 and 4n−13 edges with 4,4. Hence, M2T+++G=6×2+8×2+12×4+16×4n−13=64n−132. However, M1Pn=4n−6, M2Pn=4n−8, and FPn=8n−14, resulting that 8M1Pn+6M2Pn+FG=64n−110. Hence, we now compute the correct analytic expressions of the second Zagreb index and coindex of total transformation graphs using reformulated Zagreb indices. Moreover, the type-II semitransformation is newly introduced in this paper, and hence we also obtain the exact expressions for first and second Zagreb indices. The following theorem gives the exact expression for second Zagreb indices of first four transformations in terms of edge version of first and second Zagreb indices of the arbitrary graph.Theorem 1.
LetG be a graph with order n and size m. Then,(i)
M2T+++G=EM2G+2EM1G+8M2G+2M1G+2FG−4m(ii)
M2T++−G=EM2G+n−2EM1G+1/2n−22+2mn−2M1G+m3+m2n−2n−4−mn−22(iii)
M2T+−+G=EM2¯G−m+1EM1¯G+1/2mm+13−1/2m2−2m−11M1G−2FG(iv)
M2T−++G=EM2G+2EM1G+2nM1G+1/2nn−13−mn2−2n+5Proof.
The graphT+++G has m+n vertices and 1/2M1G+2m edges in which m edges are actual edges in G by condition (#1), 1/2M1G−m edges are produced by condition (#2) called edge adjacency relation edges (line graph edges), and 2m edges are edges produced by condition (#3) called incidence relation edges. For any vertex s∈VT+++G,(4)dT+++Gs=2dGs,ifs∈VG,dGs+2,ifs∈EG.
Therefore,(5)M2T+++G=∑st∈ET+++GdT+++GsdT+++Gt=∑st∈ET+++G∩EGdT+++GsdT+++Gt+∑st∈ET+++G∩ELGdT+++GsdT+++Gt+∑st∈ET+++G∖EG∪ELGdT+++GsdT+++Gt=∑st∈EG2dGs2dGt+∑s,t∈EGs∼tdGs+2dGt+2+∑s∈VG,t∈EGs∼t2dGsdGt+2=4M2G+∑s,t∈EGs∼tdGsdGt+∑s,t∈EGs∼t2dGs+dGt+4ELG+∑s∈VG∑x∈NGs2dGsdGs+dGx=4M2G+EM2G+2EM1G+4ELG+2∑s∈VGdGs3+2∑s∈VG∑x∈NGsdGsdGx=4M2G+EM2G+2EM1G+4M1G2−m+2FG+4M2G=EM2G+2EM1G+8M2G+2M1G+2FG−4m.
This completes the proof of assertion (i). Next, for any vertexs∈VT++−G,(6)dT++−Gs=m,ifs∈VG,dGs+n−2,ifs∈EG.
It can be seen that(7)M2T++−G=∑st∈EGdGsdGt+∑s,t∈EGs∼tdGsdGt+∑s∈VG,t∈EGs≁tdGsdGt=∑st∈EGm⋅m+∑s,t∈EGs∼tdGs+n−2dGt+n−2+∑s∈VG,t∈EGs≁tmdGt+n−2=m3+EM2G+n−2EM1G+n−22ELG+∑t∈EG∑s∈VGs≁tmdGt+n−2=m3+EM2G+n−2EM1G+n−22M1G2−m+m∑t∈EGn−2dGt+n−2=m3+EM2G+n−2EM1G+12n−22M1G−2mn−22+mn−2∑t∈EGdGt+m2n−22=m3+EM2G+n−2EM1G+12n−22M1G−2mn−22+mn−2M1G−2m+m2n−22=EM2G+n−2EM1G+12n−22+2mn−2M1G+m3+m2n−2n−4−mn−22.
To complete the proof of assertion (iii), we notice that for any vertex,s∈VT+−+G,(8)dT+−+Gs=2dGs,ifs∈VG,m+1−dGs,ifs∈EG.
As before, we can easily write that(9)M2T+−+G=∑st∈EG2dGs2dGt+∑s,t∈EGs≁tm+1−dGsm+1−dGt+∑s∈VG,t∈EGs∼t2dGsm+1−dGt=4M2G+m+12mm−12−ELG−m+1EM1¯G+EM2¯G+2∑s∈VG∑x∈NGsdGsm+3−dGs+dGx=EM2¯G−m+1EM1¯G+4M2G+12mm+13−m+12M1G+2m+3M1G−2FG−4M2G=EM2¯G−m+1EM1¯G+12mm+13−12m2−2m−11M1G−2FG.
The final assertion follows from the fact that for any vertexs∈VT−++G,(10)dT−++Gs=n−1,ifs∈VG,dGs+2,ifs∈EG.Theorem 2.
LetG be a graph with order n and size m. Then,(1)
M2T−−−G=1/2M12G−3m+n−12−82m+n−2M1G+4m+4n−22M2G+2m+2n−7FG−2EM2G−4EM1G+m+nm+n−13−12mm+n−12+8m2m+1(2)
M2T−−+G=1/2M12G−3m2+14m+12n−17M1G+4m+4n−6M2G+2m+2n−3FG−2EM2G−2n−2EM1G+m4+7m3+4m2n+11m2−2mn2+24mn−29m+n4−3n3+3n2−n(3)
M2T−+−G=1/2M12G+3m+n−12−2m2+7m−2m+12m+2n−3+m2−2m−11M1G+22m+2n−3M2G+2m+2n+1FG−2EM2¯G+2m+1EM1¯G+m+nm+n−13−3m+n−12m2+7m+m2+7m2+mm+322m+2n−3−mm+13(4)
M2T+−−G=1/2M12G+2nn−1−3m+n−12−4nM1G+22m+2n−3M2G+2m+2n−3FG−2EM2G−4EM1G+m+nm+n−13−3nn−1m+n−12+n2n−12+nn−122m+2n−3−nn−13+2mn2−2n+5Proof.
It was proved [35] that(11)M2G¯=12nn−13−3mn−12+2m2+2n−32M1G−M2G,and known that T+++G¯≅T−−−G, T++−G¯≅T−−+G, T+−+G¯≅T−+−G, and T+−−G¯≅T−++G. By Lemma 6 and Theorem 1, we can easily complete the proof.The Zagreb coindices are introduced in [36] with extensive applications in the field of chemical graph theory and widely discussed in [9, 10, 37–39]. Therefore, it will be worth finding the second Zagreb coindices of total transformations.Theorem 3.
LetG be a graph with order n and size m. Then,(1)
M2¯T+++G=1/2M12G+8m−1M1G−18M2G−5FG−2EM2G−4EM1G+8m2m+1(2)
M2¯T++−G=1/2M12G+2mn−4m−n2+2n+4M1G−2M2G−FG−2EM2G−2n−2EM1G+2m2n2−2m3−5m2n+mn2−8m(3)
M2¯T+−+G=1/2M12G−m2+14m+9M1G−2M2G+3FG−2EM2¯G+2m+1EM1¯G+10m3+40m2−10m(4)
M2¯T−++G=1/2M12G+2nn−3M1G−2M2G−FG−2EM2G−4EM1G+2mn2−2n+5(5)
M2¯T−−−G=1/22EM2G+4EM1G+m2+n2+2mn−10m−10n+13M1G−4m+n−5M2G−2m+n−3FG+4m3+8m2n−8m2+4mn2−8mn−4m(6)
M2¯T−−+G=1/22EM2G+2n−2EM1G+m2−2n2+10m+14n−11M1G−4m+n−1M2G−2m+n−1FG−2m3+2m2n2−6m2n−8m2+8mn2−30mn+20m(7)
M2¯T−+−G=1/22EM2¯G−2m+1EM1¯G+2m2−n2+2mn+4m+6n+6M1G−4m+n−1M2G−2m+n+1FG+m2n2+7mn2−32mn−2m3−16m2+26m(8)
M2¯T+−−G=1/22EM2G+4EM1G+m2+2mn−2m+n2+2n+1M1G−4m+n−1M2G−2m+n−1FG+m2n2−2mn2−m2n+4mn−10mProof.
It was shown in [35] that(12)M2¯G=2m2−12M1G−M2G,and combining the results of Lemma 6 and Theorem 1, we can finish the proof by simple mathematical calculations.The following theorem fills the gap in the literature with respect to the results found in [21, 25].Theorem 4.
LetG be a graph with order n and size m. Then,(1)
M1T2++G=M1G+FG+2M2G [25](2)
M1T2+−G=2M2G+FG+2n−7M1G+mnm+n−4mm+2n−4(3)
M1T2−+G=2M2G+FG−2m+5M1G+mm+32(4)
M1T2−−G=2M2G+FG−2m+2n−3M1G+m2n−4+mm+n−12(5)
M1¯T2++G=m+n−2M1G−2M2G−FG+2mn+2mm−1 [25](6)
M1¯T2+−G=m−n+6M1G−2M2G−FG+mnm+n−2mm+5(7)
M1¯T2−+G=m−n+6M1G−2M2G−FG+mnm+5−2mm+7(8)
M1¯T2−−G=m+n−2M1G−2M2G−FG+mnn−3+2mm+1Proof.
From the construction of type-II semitransformation, it is easily seen that for any vertexs∈VT2βγG such that s∈VG,(13)dT2βγGs=dGs,:β=+,γ=+,dGs,:β=−,γ=+,m−dGs,:β=+,γ=−,m−dGs,:β=−,γ=−.
In the same way, for any vertexs∈VT2βγG such that s∈EG,(14)dT2βγGs=dGs+2,:β=+,γ=+,m+1−dGs,:β=−,γ=+,n−2+dGs,:β=+,γ=−,m+n−3−dGs,:β=−,γ=−.
The proof follows from routine mathematical simplifications and, in addition, using the relationM1¯G=2mn−1−M1G [35].In [25], the authors have made an attempt to find the second Zagreb index of type-I semitransformation and left the calculations of type-II semitransformation due to its computational complexity. The following theorem gives the exact analytical expressions of the second Zagreb indices for type-II transformations of an arbitrary graph.Theorem 5.
LetG be a graph with order n and size m. Then,(1)
M2T2++G=EM2G+2EM1G+2M2G+2M1G+FG−4m(2)
M2T2+−G=EM2G+n−2EM1G+2M2G+1/2n2+2mn−8m−2n−4M1G+FG+m2n−42−mn−22(3)
M2T2−+G=EM2¯G−m+1EM1¯G−2M2G−1/2m2−5M1G−FG+1/2mm+13(4)
M2T2−−G=EM2¯G−m+n−3EM1¯G−1/2m+n−32+2mn−10m−2n+2M1G−2M2G−FG+1/2mm+1m+n−32+2m2n−4m+n−1(5)
M2¯T2++G=1/2M12G+4m−5M1G−3FG−2EM2G−4EM1G−6M2G+4mm+2(6)
M2¯T2+−G=1/2M12G+2mn−n2−4m+11M1G−3FG−2EM2G−2n−2EM1G−6M2G+mn2mn−9m+n+8mm−1(7)
M2¯T2−+G=1/2M12G−mm+8M1G+2M2G−2EM2¯G+2m+1EM1¯G+FG+2m3m2+8m−5(8)
M2¯T2−−G=1/2M12G−m2−n2+8m+6n−8M1G+2M2G+FG−2EM2¯G+2m+n−3EM1¯G+mnmn−m−2n+8+2m3m2+2m−5Proof.
The proof of (i)–(iv) is similar to Theorem1, and for the sake the completeness, we give the proof of (i). The graph T2++G has m+n vertices and 1/2M1G+m edges in which 1/2M1G−m edges are produced by condition (#2) called edge adjacency relation edges (line graph edges) and 2m edges are edges produced by condition (#3) called incidence relation edges. Also, for any vertex, s∈VT2++G,(15)dT2++Gs=dGs,ifs∈VG,dGs+2,ifs∈EG.
Hence,(16)M2T2++G=∑st∈ET2++GdT2++GsdT2++Gt=∑st∈ET2++G∩ELGdT2++GsdT2++Gt+∑st∈ET2++G∖ELGdT2++GsdT2++Gt=∑s,t∈EGs∼tdGs+2dGt+2+∑s∈VG,t∈EGs∼tdGsdGt+2=∑s,t∈EGs∼tdGsdGt+∑s,t∈EGs∼t2dGs+dGt+4ELG+∑s∈VG∑x∈NGsdGsdGs+dGx=∑s,t∈EGs∼tdGsdGt+∑s,t∈EGs∼t2dGs+dGt+4ELG+∑s∈VGdGs3+∑s∈VG∑x∈NGsdGsdGx=EM2G+2EM1G+4M1G2−m+FG+2M2G=EM2G+2EM1G+2M2G+2M1G+FG−4m.
To complete the remaining parts, we apply equation (12) with the help of Theorem 4.


## 3.2.F-Index of Transformation Graphs

The forgotten index and coindex of type-I semi and total transformations of graphs have been obtained [20, 28] in terms of first Zagreb, second Zagreb, vertex a-Zagreb, and (a, b)-Zagreb indices. In this section, we rewrite vertex a-Zagreb and (a, b)-Zagreb indices in terms of the sum-connectivity index. Before proceeding to this, we shall state a basic lemma.Lemma 7 (see [28]).
LetG be a connected graph of order n and size m. Then,(i)
FG¯=nn−13−FG−6mn−12+3n−1M1G(ii)
F¯G=n−1M1G−FGThe following theorem is crucial for finding the sigma index of the transformation of an arbitrary graph.Theorem 6.
LetG be a connected graph of order n and size m. Then,(1)
FT+++G=8FG+χ3G(2)
FT++−G=χ3G+3n−12FG+2M2G+3n−42M1G+mn−43+m3n(3)
FT+−+G=3m+17FG−χ3G+mm+33−3m+32M1G+6m+3M2G(4)
FT−++G=nn−13+χ3G(5)
FT−−−G=6m+n−1M2G−3m2−18m+6mn−18n+3n2+15M1G+3m+3n−11FG−χ3G+m+nm+n−13−12mm+n−12(6)
FT−−+G=3m+9FG−χ3G−3m2+18m+27M1G+6m+18M2G+m4+9m3+27m2+27m+n4−3n3+3n2−n(7)
FT−+−G=χ3G+3n−20FG+3n2−12n+12m+36M1G+6n−24M2G+m+nm+n−13−mm+33−3m+n−12m2+7m+3mm+32m+n−1(8)
FT+−−G=3m+n−1FG+6m+n−1M2G−3m+n−12M1G−χ3G+m+nm+n−13−nn−13−3mnn−1m+n−1Proof.
The proof of (i)–(iv) can be derived using the degrees of vertices from the proof of Theorem1 and the remaining parts from Lemma 7.The following theorem is an easy consequence of combining Lemma7 and Theorem 6, which will be used to compute the analytical expressions of the forgotten coindices of total transformations of an arbitrary graph.Theorem 7.
LetG be a connected graph of order n and size m. Then,(1)
F¯T+++G=4m+n−1M1G+2m+n−1M2G+m+n−9FG−χ3G(2)
F¯T++−G=m−2n+11FG−χ3G−n2−2mn+8m−14n+40M1G+2m−4n+22M2G+2m2n2−9m2n+16m2+3mn2−24mn+48m(3)
F¯T+−+G=n−2m−18FG+χ3G+m2+18m−2mn−2n+29M1G+2n−4m−20M2G+m3n−4m3+6m2n−24m2+9mn−36m(4)
F¯T−++G=m+n−1FG+2m+n−1M2G−χ3G+mn3−2mn2+mn(5)
F¯T−−−G=χ3G−2m+2n−10FG+m2+n2+2mn−10m−10n+9M1G−4m+n−1M2G+4m3+8m2n−8m2+4mn2−8mn+4m(6)
F¯T−−+G=χ3G+m2+14m−2mn−6n+33M1G−4m−2n+20M2G−2m−n+10FG+m3n−4m3+6m2n−24m2+mn3−2mn2+10mn−36(7)
F¯T−+−G=2m−2n+11M2G−χ3G−n2−2mn+16m−6n+32M1G+m−2n+19FG+4m3+m2n2+8m2+7mn2−32mn+52m(8)
F¯T+−−G=χ3G−4m+n−1M2G−2m+n−1FG+m+n−12M1G+m2n2−m2nTheorem 8 (see [20]).
LetG be a connected graph of order n and size m. Then,(1)
FT1++G=8FG+8m(2)
FT1+−G=nm3+mn−23(3)
FT1−+G=nn−13+8m(4)
FT1−−G=12m+n−1M1G−8FG+nm+n−13−12mm+n−12+mn−23(5)
F¯T1++G=4mm+n−1−8m+4m+n−1M1G−8FG(6)
F¯T1+−G=m+n−1nm2+mn−22−nm3−mn−23(7)
F¯T1−+G=m+n−14m+nn−12−nn−13−8m(8)
F¯T1−−G=8FG−8m+n−1M1G+mm+n−1n−22+m+n−12nm+n−1+4m−nm+n−13−mn−23Theorem 9.
LetG be a connected graph of order n and size m. Then,(1)
FT2++G=FG+χ3G(2)
FT2+−G=m3n−6+mn−43+χ3G+3n−13FG+6n−4M2G+3n−42+3mM1G(3)
FT2−+G=FG+mm+33−3m+32M1G+3m+3FG+2M2G−χ3G(4)
FT2−−G=m3n−6+mm+n−13+3m+3n−4FG+6m+n−1M2G+3m−3m+n−12M1G−χ3G(5)
F¯T2++G=m+n−1M1G+2M2G+m+n−2FG−χ3G(6)
F¯T2+−G=2m−4n+22M2G+2mn−n2+15n−10m−41M1G+m−2n+12FG−χ3G+m+n−1mnm+n−4mm+2n−4−m3n−6−mn−43(7)
F¯T2−+G=2n−4m−20M2G+n−2m−11FG+m2+15m−2mn−5n+32M1G+χ3G+mm+32n−4(8)
F¯T2−−G=χ3G−4m+n−1M2G+m2+2mn−4m−n+n2M1G−2m+2n−3FG+2m3+m2n2−5m2n+4m2Proof.
The proof of the theorem follows from using the degrees of vertices as given in the proof of Theorem4 and Lemma 7.


## 3.3.σ-Index of Transformation Graphs

In this section, we first derive a relation betweenσ-index of a graph and its coindex. Following this, we derive another relation between σ-coindex of a graph and σ-index of the complement graph. Finally, we list down the σ-index and coindex of semi and total transformations of graphs from the above subsections. The following theorem gives the relationship between the sigma index and its coindex.Theorem 10.
LetG be any graph with n vertices and m edges. Then,(17)σG+σ¯G=nM1G−4m2.Proof.
The proof is completed from the definitions ofσ-index and coindex as explained in the following:(18)σG+σ¯G=∑st∈EGdGs−dGt2+∑st∉EGdGs−dGt2=∑s,t⊆VGdGs−dGt2=∑s,t⊆VGdGs2+dGt2−2dGsdGt=∑s,t⊆VGdGs2+dGt2−∑s,t⊆VG2dGsdGt=FG+F¯G−2M2G−2M2¯G=n−1M1G−2M2G−22m2−M2G−M1G2=nM1G−4m2.Corollary 1.
LetG be any graph with n vertices and m edges. Then,(19)σ¯G=nM1G+2M2G−FG−4m2.The following theorem establishes interesting result that the sigma index of the complement of a graph and sigma coindex of a graph is one and the same.Theorem 11.
LetG be any graph with n vertices and m edges. Then,(20)σG¯=σ¯G.Proof.
For any vertexs∈VG, dG¯s=n−1−dGs, and we have(21)σG¯=∑st∈EG¯dG¯s−dG¯t2=∑st∉EGn−1−dGs−n−1−dGt2=∑st∉EGdGt−dGs2=σ¯G.Corollary 2.
LetG be any graph with n vertices and m edges. Then,(22)σ¯G¯=σG.The main objective of this section is the following theorem.Theorem 12.
LetG be a connected graph of order n and size m. Then,(1)
σT+++G=4FG+χ3G−2EM2G−4EM1G−16M2G−4M1G+8m(2)
σT++−G=χ3G+3n−12FG+2M2G−n−22+2mn−2−3n−42M1G−2EM2G−2n−2EM1G−2m3−2m2n−2n−4+2mn−22+mn−43+m3n(3)
σT+−+G=12FG−χ3G+3m+3χ2G−2EM2¯G+2m+1EM1¯G+m2−2m−11−3m+32M1G+mm+33−mm+13(4)
σT−++G=χ3G−2EM2G−4EM1G−4nM1G+2mn2−2n+5(5)
σT−−−G=2EM2G+4EM1G−M12G+4n−m+1M1G+2m+n+8M2G+m+n−4FG−χ3G−8m2m+1(6)
σT−−+G=2EM2G+2n−2EM1G−M12G−χ3G+12n−4m−44M1G+2m−4n+24M2G+m−2n+12FG+2m3+16m2−4m2n+2mn2−24mn+56m(7)
σT−+−G=2EM2¯G−2m+1EM1¯G−M12G+χ3G+2m2+32m−2mn+38−2nM1G+2n−4m−18M2G+n−2m−21FG+m3n−14m3+6m2n−64m2+9mn−26m(8)
σT+−−G=2EM2G+4EM1G−M12G−χ3G−2nn−3M1G+2m+2nM2G+m+nFG+n3m−4n2m+5nm−10mIn sequence to Theorems3 and 7, we have the following.Theorem 13.
LetG be a connected graph of order n and size m. Then,(1)
σ¯T+++G=2EM2G+4EM1G+2m+n+8M2G+4n−m+1M1G+m+n−4FG−χ3G−M12G−16m2−8m(2)
σ¯T++−G=2EM2G+2n−2EM1G+2m−2n+12M2G+43n−m−11M1G+m−2n+12FG−χ3G−M12G+2m3−4m2n+16m2+2mn2−24mn+56m(3)
σ¯T+−+G=2EM2¯G−2m+1EM1¯G+χ3G−M12G+2n−4m−18M2G+2m2+32m−2mn+38−2nM1G+n−2m−21FG+m3n−14m3+6m2n−64m2+9mn−26m(4)
σ¯T−++G=2EM2G+4EM1G+2m+nM2G−2nn−3M1G+m+nFG−χ3G−M12G+mnn−12−2mn2−2n+5(5)
σ¯T−−−G=σT+++G(6)
σ¯T−−+G=σT++−G(7)
σ¯T−+−G=σT+−+G(8)
σ¯T+−−G=σT−++GThe following theorems give the exact expressions of the sigma index of type-I and type-II semitransformations.Theorem 14.
LetG be graph with n and m as its order and size, respectively. Then,(1)
σT1++G=8FG−8M1G−8M2G+8m(2)
σT1+−G=nm3+mn−23−2m3−2m2n−22(3)
σT1−+G=2mn−12−8mn−1+8m(4)
σT1−−G=4m+n−1M1¯G−8FG+43m+2n−1M1G−8M2¯G+m3n−8mn2−6m2n+17mn−10m3−4m2−10m(5)
σ¯T1++G=4m+n+2M1G+8M2G−8FG−4m8m−n+2(6)
σ¯T1+−G=2m2m−2n+4+2mnn−4+8m(7)
σ¯T1−+G=mn3−8mn2+21mn−18m(8)
σ¯T1−−G=8FG−42m+n−1M1G−4m+n−1M1¯G+8M2¯G+2m2m+4n−10+2m4n2−8n+5Theorem 15.
LetG be graph with n and m as its order and size, respectively. Then,(1)
σT2++G=χ3G−2EM2G−4EM1G−4M2G−4M1G−FG+8m(2)
σT2+−G=χ3G+3n−5FG+23n−14M2G+2n2−2mn+11m−22n+52M1G−2EM2G−2n−2EM1G+m3n+mn3−6m3−2m2n2+16m2n−10mn2−32m2+40mn−56m(3)
σT2−+G=2m+1EM1¯G−2EM2¯G−2m2+18m+32M1G+3m+4FG+23m+11M2G−χ3G+6m3+24m2+26m(4)
σT2−−G=2m+n−3EM1¯G−2EM2¯G−2m2+2mn+2n2+7m+2n−8M1G+3m+3n−2FG+6m+6n−2M2G−χ3G+4m3+mn3−4mn2+8m2n+9mn−8m2−10m(5)
σ¯T2++G=2EM2G+4EM1G−3m−n−4M1G−M12G+m+n+1FG+2m+n+2M2G−χ3G−4m2−8m(6)
σ¯T2+−G=2EM2G+2n−2EM1G−M12G−χ3G+15n−6m−52M1G+2m−4n+28M2G+m−2n+15FG+2m3−4m2n+12m2+2mn2−24mn+56m(7)
σ¯T2−+G=2EM2¯G−2m+1EM1¯G−M12G+χ3G+2m2−2mn+23m−5n+32M1G−22m−n+11M2G−2m−n+12FG+m3n−10m3+6m2n−40m2+9mn−26m(8)
σ¯T2−−G=2EM2¯G−2m+n−3EM1¯G+χ3G−M12G+2m2+2mn+4m+5n−8M1G−22m+2n−1M2G−2m+n−1FG−4m3−4m2n+2mn2−8mn+10m


## 4. Results and Discussion

The various expressions for the Zagreb and sigma topological indices computed here can be extremely useful in the thermodynamic properties such as the heat of formation and entropy for the structure-property predictions of the transformation of molecular materials when combined with total and semitype transformations. Since the enumeration and construction of different structures under given specific constraints have found potential applications in drug discovery via topological indices [40, 41], it can help the chemists by reducing the number of potential drug compounds that need to be experimentally considered. We computed expressions based on the degree measures of the given graph, and hence it can be considered as an efficient technique for vibrational spectroscopic chemical analysis through the vertex partitioning and providing significant simplifications in the vibrational mode analysis. Moreover, sigma indices obtained here offer the regularity perfection of the structure.The semi and total transformation considered here provide 16 classes of new structures for the given graph based on the edge adjacency and incidence relations. Once we compute the topological indices such as Zagreb, reformulated Zagreb, forgotten, sum-connectivity, and sigma for the base graph and then using the results from Theorems1–15 , one can readily obtain the Zagreb and sigma indices for the new structures.We now present the applications of our computed results for perhydrophenalene. The molecular graph of perhydrophenaleneG is shown in Figure 4 and has 13 vertices and 15 edges. Moreover, G has 9 vertices of degree 2 and 4 vertices of degree 3. Clearly, the edge partition of G has three classes based on the degree of end vertices, namely, 2,2, 2,3, and 3,3 while the number of edges in the classes, respectively, are 6, 6, and 3.Figure 4
The molecular graph of perhydrophenaleneG.From the above data, one can easily deriveM1G=72, M2G=87, EM1G=126, EM2G=195, FG=180, EM1¯G=462, EM2¯G=624, M1¯G=288, M2¯G=327, χ2G=354, and χ3G=1782. Then, the calculations of second Zagreb and sigma indices of total and semitransformations of G are obtained from Theorems 1–15 and presented in Tables 2 and 3, respectively. These values are compared graphically and depicted in Figure 5.Table 2
Second Zagreb and sigma indices of total transformations of perhydrophenaleneG.
S. no.Total transformations ofGSecond Zagreb indexSigma index1.T+++G1587482.T++−G416523423.T+−+G1696823344.T−++G1133115845.T−−−G1553585046.T−−+G285605347.T−+−G86868125828.T+−−G998948760Table 3
Second Zagreb and sigma indices for semitransformations of perhydrophenaleneG.
S. no.Semitransformations ofGSecond Zagreb indexSigma index1.T1++G6362882.T1+−G3060026403.T1−+G979230004.T1−−G72372226085.T2++G8851926.T2+−G330693187.T2−+G1567836548.T2−−G21779415120Figure 5
(a) Total transformation of the second Zagreb and sigma indices; (b) semitransformation of second the Zagreb and sigma indices.
(a)(b)In the case of the second Zagreb index of the molecular graphG of perhydrophenalene, we infer that M2T+++G≤M2T−++G≤M2T+−+G≤M2T−−+G≤M2T++−G≤M2T−+−G≤M2T+−−G≤M2T−−−G and M2T1++G≤M2T2++G≤M2T1−+G≤M2T2−+G≤M2T1+−G≤M2T2+−G≤M2T1−−G≤M2T2−−G.On the other side, for the sigma index, we observe thatσT+++G≤σT++−G≤σT−−−G≤σT−−+G≤σT−++G≤σT+−+G≤σT+−−G≤σT−+−G and σT2++G≤σT1++G≤σT2+−G≤σT1+−G≤σT1−+G≤σT2−+G≤σT2−−G≤σT1−−G.


## 5. Conclusion

The topological characterization of graphs and their transformations has been discussed in many research papers, in particular to Zagreb indices. Unfortunately, we have noticed the study on the second Zagreb index in total transformation graphs with some technical failures such as missing out edge degree-based indices and giving incorrect expressions. In this paper, we made a detailed study and derived the exact analytic expressions by incorporating reformulated Zagreb indices. As a byproduct, we have derived the sigma index of transformation graphs effectively using the forgotten index, and in addition, we have considered all possible semitransformations. The locus of this work will be definitely useful in computing other pending topological indices which are not computed for total transformation of graphs.



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