# Computing Connection Distance Index of Derived Graphs

Authors: Faiz Farid; Muhammad Javaid; Ebenezer Bonyah
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Distance based topological indices (TIs) play a vital role in the study of various structural and chemical aspects for the molecular graphs. The first distance-based TI is used to find the boiling point of paraffin. The connection distance (CD) index is a latest developed TI that is defined as the sum of all the products of distances between pair of vertices with the sum of their respective connection numbers . In this paper, we computed CD indices of the different derived graphs (subdivision graphSG, vertex-semitotal graph RG, edge-semitotal graph QG and total graph TG obtained from the graph G under various operations of subdivision in the form of degree distance (DD) and CD indices of the basic graphs including some other algebraic expressions.


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## Body

## 1. Introduction

A topological index (TI) is a function from the set of simple graphsF to the set of real numbers that assigns a unique number to each graph G belonging to F. More importantly, it remains constant for the isomorphic graphs that is if G1≅G2 then TIG1=TIG2. Many topological indices have been introduced for the molecular graphs in the chemical graph theory to predict the certain structural and chemical properties such as vaporization, freezing point, boiling point, volume, density, weight and physicochemical properties of chemical bound [1]. Moreover, most rapidly growing fields of science such as chemistry, mathematics and information sciences provide effective research achievements in the last decade and one of the combinations of these subjects is called by cheminformatics which studies two relationships techniques quantitative structures property relationships (QSPR) and quantitative structure activity relationships (QSAR) for the different molecular structures, see [2–4].Wiener (1947) [5] was first scientist who laid the foundation of chemical graph theory. In chemical graph theory, atoms and bounds are taken as vertices and edges. He discovered through his research work that there is a close correlation between the sums of the distances among pairs of vertices and the boiling points of paraffine. Almost, after passage of three quarters of the 20th century (1972) [6] discovered degree-based indices, (First and Second Zagreb indices), which were utilized to calculate the total π-electron energy of molecules. After these developments, many TIs were introduced which are found to be highly useful for the study of different physicochemical properties of chemical compounds. Degree distance index [7] and Gutman index [8] are the most important distance-degree based TIs. Ali and Trinajstic [9] (2018) restudied the first Zagreb connection index ZC1, second Zagreb connection index ZC2 and the modified first Zagreb connection index ZC1∗. Recently, Javaid et al. [10] defined a new distance-based TI called by connection distance index as the sum of all the products of distances between pair of vertices with the sum of their respective connection numbers.Yan et al. [11] defined the new graphs called as subdivided graph SG, vertex-total graph RG, edge-total graph QG and total graph TG with the help of the subdivision-related operations S, R, Q and T respectively. Xu et al. [12] have determined the degree distance index of the derived graphs. Bahadur et al. [13] have determined Gutman index of derived graphs. In particular, on different families of graphs for the result about degree based TIs, see [14, 15] (line graph), [16, 17] (sum-graphs), [18, 19] (derived graphs), [20, 21] (extremal graphs), [22, 23] (generalized sum graphs), [24] (nanotube) and distance based TIs, see [25–27].In this paper, firstly, we have determined the exact values of connection numbers of old (black) vertices and new (white) for derived graphSG and bounded values of connection numbers for C3,C4− graph of RG,QG and TG were determined. Then, finally, bounded and exact values of connection distance (CD) of derived graphs SG,RG,QG,TG are determined by Theorems and Corollaries. Section 2 consists of applications, Section 3 comprises some related definitions, Section 4 covers the main results of connection distance index (CD) for the derived graphs and Section 5 sum up our findings.


## 2. Applications

From the last two decades, the worth investing problem of finding out the physicochemical properties of the molecular structures is attracting the attention of chemists and mathematicians continuously. The TIs have been used to: predict the solubility of fullereneC60 [28], find the ionic liquids densities [1], optimize QSPR models (Simplex optimisation of generalized TIs) [29], manufacture the anticancer drugs [30, 31], calculate molecular van der Waals areas or volumes [32] and measure the vaporisation, sublimation, formation & combustion for the monocarboxylic acids C2H4O2−C20H40O2 [4]. Recently, International Academy of Mathematical Chemistry checked thirteen physicochemical properties of octane isomers (heat capacity at T constant, heat capacity at P constant, density, boiling point, entropy, acentric factor, enthalpy of formation, octanol-water partition coefficient, enthalpy of vaporisation, standard enthalpy of formation, molar volume and total surface tension) with the help of the connection number-based TIs and declared that the chemical capability of the Zagreb connection indices is better than the ordinary Zagreb indices for the entropy and acentric factor of the octane isomers, for detail see [9]. In addition, Javaid et al. [33] presents comparison of correlation coefficients between different TIs and eleven physicochemical properties of octane isomers (boiling point, heat capacity, entropy, density,mean radius, change in heat of vaporization, standard heat of formation, acentric factor, enthalpy of vaporization and standard enthalpy of vaporization) which shows that Gutman connection index is a very useful TI for the prediction of entropy, acentric factor, enthalpy of vaporization and standard enthalpy of vaporization.


## 3. Preliminaries

Throughout in this research paper, we take a simple and connected graphsG with VG=ui:1≤i≤n and EG=ei:1≤i≤m such that VG=n and EG=m. For more basic notations of graphs, see [34, 35]. Now, we define some most frequent used definition as follows•
The number of vertices at distance one from a vertexui is called its degree and it is denoted by δui.•
Degree of an edgeei=ujuk is given by δei=δuj+δuk−2, where 1≤i≤m for some 1≤j,k≤n.•
The distance between two verticesui,uj∈VG is denoted by dui,uj and defined as the length of the shortest path between both the vertices ui and uj for 1≤i,j≤n.•
The distance between two edgesei=upuq and ej=urus is defined as dGei,ej=mindGup,ur,dGup,us,dGuq,ur,dGuq,us, where 1≤i,j≤m and 1≤p,q,r,s≤n.•
The distance between one vertexui and one edge ej=upuq is defined as  dGui,ej=mindGui,up,dGui,uq, where 1≤j≤m and 1≤i,p,q≤n.•
The number of vertices at distance two from a vertexui is called its connection number of ui and it is denoted by τui for 1≤i≤n .More details about aforesaid definitions can be obtained from [9, 36, 37].Some important and related TIs are the following:Definition 1 (see [5]).
LetG be a connected graph of order n, then its Wiener index is(1)WG=12∑ui,uj∈VGdGui,uj,where 1≤i,j≤n.Definition 2 (see [6]).
LetG be a connected graph of order n, then first and second Zagreb index are defined as(2)M1G=∑uiuj∈EGδGui+δGuj=∑ui∈VGδGui2.
(3)M2G=∑uiuj∈EGδGuiδGuj,where 1≤i,j≤n.Definition 3 (see [38]).
LetG be a connected graph of order n, then first and second Zagreb coindex are defined as(4)M1G¯=∑uiuj∉EGδGui+δGuj.
(5)M2G¯=∑uiuj∉EGδGuiδGuj,where 1≤i,j≤n.Definition 4 (see [6]).
LetG be a connected graph of order n and size m, then edge version of Wiener index is defined as(6)WeG=∑ei,ej⊆EGdGei,ej+1,where 1≤i,j≤m.Definition 5 (see [35]).
LetG be a connected graph of order n, then the degree distance index of G is(7)DDG=12∑ui,uj∈VGdui,ujδui+δuj,where 1≤i,j≤n.Definition 6 (see [12]).
LetG be a connected graph of order n and size m, then the edge version of degree distance index of G is(8)DDeG=∑ei,ej⊆EGdeei,ej+1δei+δej,where 1≤i,j≤m.Definition 7 (see [8]).
LetG be a connected graph, then the Gutman index of G is(9)GutG=12∑ui,uj∈VGdui,ujδuiδuj,where 1≤i,j≤n.Definition 8 (see [10]).
Connection Distance index of a graph G (denoted byCDG) is defined as(10)CDG=∑ui,uj⊆VGdGui,ujτGui+τGuj,where 1≤i,j≤n or(11)CDG=12∑u,v∈VGdu,vτu+τv,Definition 9 (see [10]).
Gutman Connection index of a graph G (denoted byGCG) is defines as(12)GCG=∑ui,uj⊆VGdGui,ujτGuiτGuj,where 1≤i,j≤n or(13)GCG=12∑u,v∈VGdu,vτuτv.
Yan et al. [11] defined the four operations S, R, Q and T on the graph G and obtained the four new graphs from G as follows:•
SG is obtained from G if a new vertex wi is inserted in every edge ei of G. The already existing vertices ui of G are named as old or black vertices while new vertices wi are also named new or white vertices.•
RG is formed by assigning a new vertex wi corresponding to each edge ei of G and this new vertex wi is connected with the end vertices of the respective edge ei.•
QG is formed from SG if two white vertices wi and wj are further joined together when corresponding edges ei and ej have one common end vertex.•
TG is formed from RG if two white vertices wi and wj are further joined together when corresponding edges ei and ej have one common end vertex.


## 4. Relation Between Connection Numbers and Degrees

In order to determine connection numbers or bounded values of connection numbers of derived graphsSG, RG, QG and TG, we will develop Lemma 1 to Lemma 4. Before to the proofs of these results, in particular we consider Figure 1 and Figure 2 to present some trees, cycles and graphs consisting of cycles of order 3 and 4.Figure 1
Some trees and cycles.Figure 2
Some simple and connectedC3,C4− graphs.Lemma 1.
For a simple and connected graphG (i) τSGui=δGui and (ii) τSGwi=δGuj+δGuk−2=δei where wi is a new vertex corresponding to edge ei=ujuk.Proof 1.
On applying S-operation, subdivision graphSG, each vertex ui is connected with δGui=di new (white) vertices. So, δGui=di new (white) vertices wi are at distance one from old vertices ui. Further di old vertices uj are at distance two from ui in SG (which are at distance one in G)
Also, each new vertex (white)wi corresponds to each edge ei=ujuk is at distance one from uj and uk. So, number of vertices at distance 2 from wi are δGuj−1+δGuk−1 (see Figure 3 and Figure 4). Therefore τSGwi=δGuj+δGuk−2=δGei.Figure 3
S-operation of some trees and cycles.Figure 4
S-operation of simple and connectedC3,C4− graphs.Lemma 2.
LetG be a simple and connected graph(a)
IfG is a C3,C4− free graph, then(i)
τRGui=2τGui and(ii)
τRGwi=2δGuj+δGuk−4=2δGei.(b)
IfG is a C3,C4− graph, then(i)
τRGui⩽2τGui+r, where r=maxri and ri are number of C3 and C4 cycles connected with ui in G.(ii)
τRGwi⩽2δGuj+δGuk−4−s=2δGei−s, where s=maxsi and si are number of C3 cycles connected with ui in G.Proof 2.
On applyingR−operation, a new vertex wi is joined with the corresponding vertices of edge ei=ujuk.
Case (a)
IfG is a C3,C4− free graph, then the degree of each vertex ui in RG is twice the degree ui in G i.e. δRGui=2δGui (see Figure 5) . Then their connection number of each vertex ui in RG is twice the connection number ui in G i.e. τRGui=2τGui. Also uj and uk are at distance one from wi, but in R(G), degree of uj and uk are also doubled, then number of vertices at distance 2 of wi are 2δGuj−1+δGuk−1=2δGuj+δGuk−42δGei = 2δGei. Also for n≥5, δCnui=2, τCnui=2, τRCnui=4, τRCnwi=4, ⇒τRCnui=2τCnui and τRCnwi=2δCnuj+δCnuk−4 = 2δCnei, i=1,2…n.
Case (b)
IfG is a C3,C4− graph, then consider δC3ui=2,δC3ui=2,τC3ui=0, τRC3ui=1, τRC3wi=3⇒τRC3ui=2τC3ui+1 and τRC3wi=2δC3uj+δC3uk−5,= 2δGei−1i=1,2,3.
δC4ui=2,τC4ui=1, τRC4ui=3, τRC4wi=4, ⇒τRC4ui=2τC4ui+1 and τRC4wi=2δC4uj+δC4uk−4 = 2δC4ei, i=1,2,3,4.
ConsiderRG1 graph in which two C3 are connected by a common vertex u3 as shown in Figure 6, then δG1ui=2, τG1ui=2,τRG1ui=5⇒τRG1ui=2τG1ui+1 for i=1,2,4,5 and δG1u3=4, τG1u3=0,τRG1u3=2⇒τRG1u3=2τG1u3+2 as u3 lies in two C3 graphs. Also τRG1wi=3⇒τRG1wi=2δG1uj+δG1uk−5 = 2δGei−1 for i=1,4 and τRG1wi=7⇒τRG1wi=2δG1uj+δG1uk−5 = 2δGei−1, for i=2,3,5,6.
ConsiderRG2 graph in which three C3 are connected by a common vertex u3 as shown in Figure 6, then δG2ui=2, τG2ui=4,τRG2ui=9⇒τRG2ui=2τG2ui+1 for i=1,2,4,5,6,7 and δG2u3=6, τG2u3=0,τRG2u3=3⇒τRG2u3=2τG2u3+3 as u3 lies in three C3 graphs. Also τRG2wi=3⇒τRG1wi=2δG2uj+δG2uk−5 = 2δGei−1 for i=1,4,7 and τRG1wi=11⇒τRG1wi=2δG2xi+δG2yi−5 = 2δGei−1, for i=2,3,5,6,8,9.
ConsiderRG3 in which one C3 and one C4 are connected by a common vertex u3 as shown in Figure 6, then δG3ui=2 for i=1,2,4,5,6, τG3ui=2 for i=1,2,τG3ui=3 for i=4,6,τG3ui=1 for i=3,5,τRG3ui=5=2τG3ui+1, for i=1,2τRG3u3=4=2τG3u3+2, as u3 lies in one C3 and one C4 graphs, τRG3ui=7=2τG3ui+1, for i=4,6τRG3u5=3=2τG3u5+1, Also τRG3w1=3=2δG3u1+δG3u2−5 = 2δGei−1, τRG3w2=7=2δG3u2+δG3u3−5 = 2δGei−1, τRG3w3=8=2δG3u3+δG3u4−4 = 2δGe3, τRG3w4=4=2δG3u4+δG3u5−4 = 2δGe4.
ConsiderRG4 graph in which two C4 graphs and one path are connected by vertex u3 as shown in Figure 6. As δG4ui=2 for i=1,2,4,6,8, δG1u3=5 and δG4u5=1, τG4u1=1, τG4u2=4, τG4u3=3, τG4u4=4,τG4u5=1, τRG4u1=3=2τG4u1+1,τRG4u2=9=2τG4u2+1, τRG4u3=8=2τG4u3+2,τRG4u4=8=2τG4u1,τRG4u5=2=2τG4u5, τRG4w1=4=2δG4u1+δG4u2−4 = 2δGe1, τRG4w2=10=2δG4u2+δG4u3−4 = 2δGe2,τRG4w3=10=2δG4u3+δG4u4−4 = 2δGe3τRG4w4=2=2δG4u4+δG4u5−4 = 2δG4e4.
Consider a graphRG5, in which one edge u2u3 is common in two C3−graphs as shown in Figure 6. δG5ui=2 and τG5ui=1, τRG5u1=4=2τG5u1+2 as Nu1∈ two C3 graphs, for i=1,4, δG5ui=3 and τG5ui=0,τRG5u1=2=2τG5u1+2 as Nui∈C3 graphs, for i=2,3, τRG4w1=5=2δG4u1+δG4u2−5 = 2δG5e1, similarly we can get w2,w3 and w4, τRG4w5=5=2δG4u2+δG4u4−6 = 2δG5e5−2, as u2u3∈ two C3 graph.
Now consider a graphRG6 in which one edge u2u3 is common in three C3−graphs as shown in Figure 6, δG6ui=2 and τG6ui=2, τRG6u1=7=2τG6u1+3 as Nu1∈ three C3 graphs, for i=1,4,5, δG6ui=4 and τG6ui=0, τRG6u1=3=2τG6u1+3 as Nu1∈ three C3 graphs, for i=2,3, τRG4w1=7=2δG6u1+δG6u2−5 = 2δG6e1. Similarly, we can get w3,w4,w5,w6, and w7, τRG4w2=9=2δG4u2+δG4u4−7 = 2δG4e2−3, as u2u3∈ three C3 graph.
Now, we close our discussion by takingr=maxri and s=maxsi, then upper bound for connection number is taken is taken as τRGui=2τGui+r and τRGwi=2δGxi+δGyi−4=2δGei. And lower bound for connection number is taken as τRGui=2τGui and τRGwi=2δGxi+δGyi−4−s=2δGei−s.Figure 5
R-operation of trees and cycles.Figure 6
R-operation of simple and connectedC3,C4− graphs.Lemma 3.
For a simple and connected graphG,(a)
IfG is a C3,C4− free graph, then(i)
τQGui=δGui+τGui and(ii)
τQGwi=τGxi+τGyi.(b)
IfG is a C3,C4− graph, then(i)
τQGui⩽δGui+τGui+r where r=maxri and ri is the number of C3 and C4 cycles connected with vertex ui .(ii)
τQGwi⩽τGuj+τGuk+s where s=maxsi and si is the number of C3 cycles in graph G connected with edge ei.Proof 3.
Case (a)
IfG is a C3,C4− free graph, On applying first S-operation, the connection number of ui becomes equal to δui. Then for Q-operation, further two new (white) vertices of SG are also joined if their corresponding edges have a common vertex between them. So connection number of ui in QG becomes equal to the sum of δui and τGui.
(For explanation from Figure7, first consider a tree T with nine vertices in which δTui=1 for i=1,7,8,9, δTui=2 for i=2,3,4,5. δTu6=4,τGui=1i=1,2,6,τTui=3i=3,4, τTu5=4,τGui=3i=7,8,9. Then τQTu1=2=δTu1+τTu1,τQTu2=3=δTu2+τTu2, τQTu3=4=δTu3+τTu3=τQTu4,τQTu5=6=δTu5+τTu5, τQTu6=5=δTu6+τGu6,τQTu7=4=δTu7+τTu7=τQTu8=τQTu9, τQTw1=2=τTu1+τTu2,τQTw2=3=τTu2+τTu3,τQTw3=τTu3+τTu4, τQTw4=6=τTu4+τTu5,τQTw5=5=τTu5+τTu6,τQTw6=4=τTu6+τTu7=τQTw7=τQTw8, here we conclude τQTui=δTui+τTui and τQGTi=τGuj+τTuk).
Also forn≥5, δCnui=2, τCnui=2, τQCnui=4, τqCnwi=4, ⇒τQCnui=δCnui+τCnui and τQGwi=τGuj+τGuk.
Case(b)
ConsiderδC3ui=2,τC3ui=0, τQC3ui=3, τQC3wi=1⇒τQC3ui=δC3ui+τC3ui+1 and τQC3wi=τGuj+τGuk+1, i=1,2,3.
δC4ui=2,τC4ui=1, τQC4ui=4, τQC4wi=3,⇒τQC4ui=δC4ui+τC4ui+1 and τQGwi=τGuj+τGuk, i=1,2,3,4.
ConsiderQG1 graph in which two C3 are connected by a common vertex u3 as shown in Figure 8, then δG1ui=2, τG1ui=2,τQG1ui=5⇒τQG1ui=δG1ui+τG1ui+1 for i=1,2,4,5 and δG1u3=4, τG1u3=0,τQG1u3=6⇒τQG1u3=δG1ui+τG1u3+2 as u3 lies in two C3 graphs. Also τQG1wi=3⇒τQG1wi≠τGuj+τGuk, but τQG1wi=δG1u3−1 when u1 and u2 is of same degree and u3 is their third vertex in C3, Also τQG1wi⋖τGuj+τGuk+1 for i=1,4 and τQG1wi=4⇒τQG1wi=τGxi+τGyi+2, for i=2,3,5,6.
ConsiderQG2 graph in which three C3 are connected by a common vertex u3 as shown in Figure 8, then δG2ui=2, τG2ui=4,τQG2ui=7⇒τQG2ui=δG1ui+τG2ui+1 for i=1,2,4,5,6,7 and δG2u3=6, τG2u3=0,τQG2u3=9⇒τQG2u3=δG1u3+τG2u3+3 as u3 lies in three C3 graphs. Also τQG2wi=5⇒τQG1wi≠τGuj+τGuk for i=1,4,7 but τQG1wi=δG1u3−1 when u1 and u2 is of same degree and u3 is their third vertex in C3, Also τQG1wi⋖τGxi+τGyi+1 for i=1,4,7 and τQG1wi=7⇒τQG1wi=τGuj+τGuk+3, for i=2,3,5,6,8,9.
ConsiderQG3 graph in which one C3 and one C4 are connected by a common vertex u3 as shown in Figure 8, then δG3ui=2 for i=1,2,4,5,6, τG3ui=2 for i=1,2,τG3ui=3 for i=4,6,τG3ui=1 for i=3,τQG3u1=5=δG1ui+τG3u1+1, τQG3u3=7=δG1ui+τG3u3+2, as u3 lies in one C3 and one C4 graphs, τQG3u4=6=δG1u4+τG3u4+1,τQG3u5=4=δG1u5+τG3u5+1, . Also τQG3w1=3=δG1u3−1 when u1 and u2 is of same degree and u3 is their third vertex in C3, Also τQG1w1⋖τGu1+τGu2+1, τQG3w2=5=τGu2+τGu3+2, τQG3w3=6=τGu4+τGu3+2,τQG3w4=5=τGu4+τGu5+1.
ConsiderQG4 graph in which two C4 graphs and one path are connected by vertex u3 as shown in Figure 8. As δG4ui=2 for i=1,2,4, δG1u3=5 and δG4u5=1, τG4u1=1, τG4u2=4, τG4u3=3, τG4u4=4,τG4u5=1, τQG4u1=4=δG4u1+τG4u1+1,τQG4u2=7=δG4u2+τG4u2+1, τQG4u3=10=δG4u3+τG4u3+2,τQG4u4=6=δG4u4+τG4u4, τQG4u5=4=δG4u5+τG4u5, τQG4w1=6=τGu1+τGu2+1, τQG4w2=9=τGu2+τGu3+2,τQG4w3=9=τGu3+τGu4+2, τQG4w4=6=τGu4+τGu5.
Now we consider a graphQG5, in which one edge u2u4 is common in two C3−graphs as shown in Figure 8 . δG5ui=2 and τG5ui=1, τQG5u1=5=δG4u1+τG5u1+2 as Nu1∈ two C3 graphs, for i=1,3, δG5ui=3 and τG5ui=0,τQG5ui=5=δG4ui+τG5ui+2 as Nui∈C3 graphs, for i=2,4, τQG4w1=3=τGu1+τGu2+2. Similarly, we can get w2,w3 and w4, τQG4w5=τGu2+τGu4+2, as u2u4∈ two C3 graph.
Now we consider a graphQG6 in which one edge u2u4 is common in three C3−graphs as shown in Figure 8, δG6ui=2 and τG6ui=2, τQG6u1=7=δG4u1+τG6u1+3 as Nu1∈ three C3 graphs, for i=1,3,5, δG6ui=4 and τG6ui=0, τQG6ui=3=δG4ui+τG5ui+3 as Nui∈ three C3 graphs, for i=2,4, τQG4w1=τGu1+τGu2+2 =6. Similarly, we can get w2,w3,w4,w6, and w7, τRG4w5=3=τGu2+τGu4+3, as u2u4∈ three C3 graph.
Now, we close our discussion by taking,r=maxri and s=maxsi, τQGui⩽δGui+τGui+r and τQGwi⩽τGuj+τGuk+s.Figure 7
Q-operation of trees and cycles.Figure 8
Q-operation of simple and connectedC3,C4− graphs.Lemma 4.
For a simple, connected graphG,(a)
IfG is a C3,C4− free graph, then(i)
τTGui=2τGui and(ii)
τTGwi=τGuj+τGuk.(b)
IfG is a C3,C4− graph, then(i)
τTGui⩽2τGui+r where r=maxri and vertex ui is connected with ri number of C3 and C4 cycles and(ii)
τTGwi⩽τGuj+τGuk+s where s=maxsi and edge ei is connected with the si number of C3 cycles in graph G.Proof 4.
After applying T-operation onG, we get m new vertices wi corresponding to each edge ei=ujuk=xiyi for uj=xi, uk=yi.
Case (a)
IfG is a C3,C4− free graph, on applying first R-operation, the connection number of ui becomes equal to τui. Then for T-operation, further two new (white) vertices of RG are also joined if their corresponding edges have a common vertex between them. So connection number of ui in TG becomes equal to the twice of τui.
(For explanation from Figure9, first consider δP6ui=1 for i=1,6, δP6ui=2 for i=2,3,4,5,τP6ui=1i=1,2,5,6,τP6u3=2=τP6u4,τTP6ui=2 for i=1,2,5,6,τTP6u3=4=τTP6u4τTP6w1=2=τP6u1+τP6u2=τTGw5, τTP6w2=3=τP6u2+τP6u3=τTGw4, τTP6w3=4=τP6u3+τP6u4 here we conclude τTPnui=2τPnui and τPnwi=τPnuj+τPnuk
Also forn≥5,δCnui=2, τCnui=2, τTCnui=4, τTCnwi=4, ⇒τTCnui=2τCnui and τTGwi=τGuj+τGuk).
Case (b)
Now considerδC3ui=2,τC3ui=0, τTC3ui=1, τTC3wi=1⇒τTC3ui=2τC3ui+1 and τQC3wi=τGuj+τGuk+1, i=1,2,3.
δC4ui=2,τC4ui=1, τTC4ui=3, τQC4wi=3,⇒τQC4ui=2τC4ui+1 and τTC4wi=τGuj+τC4uk+1, i=1,2,3,4.
ConsiderTG1 in which two C3 are connected by a common vertex u3 as shown in figure 10, then δG1ui=2, τG1ui=2,τQG1ui=5⇒τTG1ui=2τG1ui+1 for i=1,2,4,5 and δG1u3=4, τG1u3=0,τTG1u3=2⇒τTG1u3=2τG1u3+2 as u3 lies in two C3 cycles. Also τTG1wi=3⇒τTG1wi≠τGuj+τGuk, but τTG1wi=δG1u3−1 when u1 and u2 is of same degree 2 and u3 is their third vertex in C3, Also τQG1wi⋖τGuj+τGuk+1 for i=1,4 and τTG1wi=4⇒τTG1wi=τGxi+τGyi+2, for i=2,3,5,6.
ConsiderTG2 in which three C3 are connected by a common vertex u3 as shown in Figure 10, then δG2ui=2, τG2ui=4,τTG2ui=9⟶τTG2ui=2τG2ui+1 for i=1,2,4,5,6,7 and δG2u3=6, τG2u3=0,τTG2u3=3⇒τTG2u3=2τG2u3+3 as u3 lies in three C3 graphs. Also τTG2wi=5⇒τTG1wi≠τGuj+τGuk for i=1,4,7 but τTG1wi=δG1u3−1 when u1 and u2 is of same degree 2 and u3 is their third vertex in C3,moreover τTG1wi⋖τG2xi+τG2yi+1 for i=1,4,7 and τTG1wi=7⇒τTG1wi=τGuj+τGuk+3, for i=2,3,5,6,8,9.
ConsiderTG3 in which one C3 and one C4 are connected by a common vertex u3 as shown in Figure 10, then δG3ui=2 for i=1,2,4,5,6, τG3ui=2 for i=1,2,τG3ui=3 for i=4,6,τG3u3=1,τTG3u1=5=2τG3u1+1, τTG3u3=4=2τG3u3+2, as u3 lies in one C3 and one C4 graphs, τTG3u4=7=2τG3u4+1, τTG3u5=3=2τG3u5+1 . Also τTG3w1=3=δG1u3−1 when u1 and u2 are of same degree and u3 is their third vertex in C3, Also τTG1w1〈τG3u1+τG3u2+1, τTG3w2=5=τG3u2+τG3u3+2, τTG3w3=6=τGu4+τGu3+2,τQG3w4=5=τG3u4+τG3u5+1.
ConsiderG4 graph in which two C4 graphs and one path are connected by vertex u3 as shown in Figure 10. As δG4ui=2 for i=1,2,4, δG4u3=5 and δG4u5=1, τG4u1=1, τG4u2=4, τG4u3=3, τG4u4=4,τG4u5=1, τTG4u1=3=2τG4u1+1,τTG4u2=9=2τG4u2+1, τTG4u3=8=2τG4u3+2,τTG4u4=8=2τG4u4, τTG4u5=2=2τG4u5, τTG4w1=6=τG4u1+τG4u2+1, τTG4w2=9=τG4u2+τG4u3+2,τTG4w3=9=τGu3+τGu4+2, τTG4w4=5=τG4u4+τG4u5.
Now we consider a graphG5, in which one edge u2u4 is common in two C3−graphs . δG5ui=2 and τG5ui=1, τTG5u1=3=2τG5u1+1, for i=1,3, δG5ui=3 and τG5ui=0,τTG5ui=2=2τG5ui+2 as ui∈ two C3 cycles, for i=2,4, τTG5w1=3=τG5u1+τG5u2+2. Similarly, we can get w2,w3 and w4, τTG4w5=2=τG5u2+τG5u4+2, as u2u4∈ two C3 graph.
Now we consider a graphG6 in which one edge u2u3 is common in three C3−graphs as shown in Figure 10, δG6ui=2 and τG6ui=2, τTG6u1=7=2τG6u1+3 as Nu1∈ three C3 graphs, for i=1,4,5, δG6ui=4 and τG6ui=0,τTG6ui=3=2τG5ui+3 as Nui∈ three C3 graphs, for i=2,4, τTG4w1=5=τG6u2+τG6u4+1. Similarly, we can get w2,w3,w4,w6, and w7, τTG4w5=3=τG6u2+τG6u4+3, as u2u4∈ three C3 graph.
Now, we close our discussion by takingr=maxri and s=maxsi, τTGui⩽δGui+τGui+r and τTGwi⩽τGuj+τGuk+s.Figure 9
T-operation of trees and cycles.Figure 10
T-operation of simple and connectedC3,C4− graphs.


## 5. Connection Distance Indices of derived graphs

Now, we will determine degree connection sum of derived graphsSG, RG, QG and SG by the following theorems.Theorem 1.
IfSG is subdivision graph of G, then(14)CDSG=2DDG+2DDeG+12nM1+mm−n+∑i=1n∑j=1mδGui+δGejdGui,ej.Proof 5.
Bylemma 5.
τSGui=δGui and τSGwi=δGuj+δGuk−2=τSGei
Also(15)dSui,uj=2dGui,uj,dSwi,wj=2dGei,ej+1,dSui,wj=2dGui,ei+1,CDSG=∑u,v⊆VGdSGui,ujτSGui+τSGuj,=12∑i=1,j=1nτSui+τSujdSui,uj+12∑i=1,j=1mτSwi+τSwjdSwi,wj,+12∑i=1n∑j=1mτSui+τSwjdSui,wj,=12∑i=1,j=1nδGui+δGuj2dGui,uj+12∑i=1,j=1mδGei+δGej2dGei,ej+1,+12∑i=1n∑j=1mδGui+δGej2dGui,ej+1,=2DDG+2DDeG+12∑i=1n∑j=1mδGui+δGej+∑i=1n∑j=1mδGui+δGejdGui,ej,=2DDG+2DDeG+12∑j=1m∑i=1nδGui+12∑i=1n∑j=1mδGej,+∑i=1n∑j=1mδGui+δGejdGui,ej,=2DDG+2DDeG+m2+12nM1−2m+∑i=1n∑j=1mδGui+δGejdGui,ej,=2DDG+2DDeG+12nM1+mm−n+∑i=1n∑j=1mδGui+δGejdGui,ej.Theorem 2.
IfG be a C3,C4− graph and RG is an vertex-semitotal graph of G, then(16)aCDRG≤2CDG+2rWG+2DDeG+2m+nM1−2m+∑i=1n∑j=1mτGuidGui,ej++∑i=1n∑j=1mδGuidGui,ej+r2∑i=1n∑j=1mdGui,ej+m∑i=1nτGui+rmn2,bCDRG≥2CDG+2DDeG+2m+nM1−2m+∑i=1n∑j=1mτGuidGui,ej−s∑i,j=1mdGei,ei+∑i=1n∑j=1mδGuidGui,ej−s2∑i=1n∑j=1mdGui,ej+m∑i=1nτGui−sm2m+n2.Proof 6.
(a)
For upper bounds, we take by lemma 2,τRGui=2τGui+r and τRGwi=δGuj+δGuk−2=δGei.
AlsodRui,uj=dGui,uj, for ui,uj∈VG. . dRwi,wj=dGei,ej+2, for ei,ej∈EG. dRui,wj=dGui,ej+1, for ui∈VG and ej∈EG.(17)CDRG=∑ui,uj⊆VGdRGui,ujτRGui+τRGuj,=12∑i=1,j=1nτRui+τRujdRui,uj+12∑i=1,j=1mτRwi+τRwjdRwi,wj+12∑i=1n∑j=1mτRui+τRwjdRui,wj′≤∑12∑i,j=1n2τGui+r+2τGuj+rdGui,uj+12∑i,j=1m2δGei+2δGejdGei,ej+2+12∑i=1n∑j=1m2τGui+r+2δGejdGui,ej+1,=∑i,j=1nτGui+τGujdGui,uj+r∑i,j=1ndGui,uj+∑i,j=1mδGei+δGejdGei,ej+1+∑i,j=1mδGei+δGej+∑i=1n∑j=1mτGuidGui,ej+∑i=1n∑j=1mδGuidGui,ej+r2∑i=1n∑j=1mdGui,ej+∑i=1n∑j=1mτGui+r2∑i=1n∑j=1m+∑i=1n∑j=1mδGej.=2CDG+2rWG+2DDeG+2mM1−2m+∑i=1n∑j=1mτGuidGui,ej++∑i=1n∑j=1mδGuidGui,ej+r2∑i=1n∑j=1mdGui,ej+m∑i=1nτGui+rmn2+nM1−2m.=2CDG+2rWG+2DDeG+2m+nM1−2m+∑i=1n∑j=1mτGuidGui,ej++∑i=1n∑j=1mδGuidGui,ej+r2∑i=1n∑j=1mdGui,ej+m∑i=1nτGui+rmn2.
For lower bounds, bylemma 6.
τRGui=2τGui and τRGwi=δGuj+δGuk−2−s=δGei−s.(18)CDRG=∑ui,uj⊆VGdRGui,ujτRGui+τRGuj=12∑i=1,j=1nτRui+τRujdRui,uj+12∑i=1,j=1mτRwi+τRwjdRwi,wj+12∑i=1n∑j=1mτRui+τRwjdRui,wj≥12∑i,j=1n2τGui+2τGujdGui,uj+12∑i,j=1m2δGei−s+2δGej−sdGei,ej+2+12∑i=1n∑j=1m2τGui+2δGej−sdGui,ej+1.=∑i,j=1nτGui+τGujdGui,uj+∑i,j=1mδGei+δGejdGei,ej+1+∑i,j=1mδGei+δGej−s∑i,j=1mdGei,ej−2s∑i,j=1m+∑i=1n∑j=1mτGuidGui,ej.+∑i=1n∑j=1mδGuidGui,ej−s2∑i=1n∑j=1mdGui,ej+∑i=1n∑j=1mτGui−s2∑i=1n∑j=1m+∑i=1n∑j=1mδGej=2CDG+2DDeG+2mM1−2m−s∑i,j=1mdGei,ei−2sm2+∑i=1n∑j=1mτGuidGui,ej+∑i=1n∑j=1mδGuidGui,ej−s2∑i=1n∑j=1mdGui,ej+m∑i=1nτGui−smn2+nM1−2m.=2CDG+2DDeG+2m+nM1−2m+∑i=1n∑j=1mτGuidGui,ej−sm2m+n2−s∑i,j=1mdGei,ei+∑i=1n∑j=1mδGuidGui,ej−s2∑i=1n∑j=1mdGui,ej+m∑i=1nτGui.
IfG is C3,C4− free graph, then connection distance index (CD)of RG can also be determined by taking r=0 and s=0 in Theorem 2.Corollary 1.
IfG be a C3,C4− free graph,then(19)CDRG=2CDG+2DDeG+2m+nM1−2m+∑i=1n∑j=1mτGuidGui,ej+∑i=1n∑j=1mδGuidGui,ej+m∑i=1nτGui.Theorem 3.
IfQG is an edge-semitotal graph of G, then(20)CDQG=DDG+CDG+2rWG+2mn+n∑i=1nτGui+rn2+2sWeG+12∑i=1,j=1mτGxi+τGyi+τGxj+τGyjdGei,ej+1+12∑i=1n∑j=1mδGui+τGuidGui,ej+1+r+s2∑i=1n∑j=1mdGui,ej+1+12∑i=1n∑j=1mτGxj+τGyjdGei,ej+1.Proof 7.
Bylemma 7.
τQGui⩽δGui+τGui+r and τQGwi⩽τGuj+τGuk+s.
To avoid confusion, we take edgeei=xiyi=ujuk for some j and k, then τQGwi⩽τGxi+τGyi+s. Also(21)dQui,uj=dGui,uj+1,dQwi,wj=dGei,ej+1,dQui,wj=dGui,ei+1,CDQG=∑ui,uj⊆VQdQGui,ujτQGui+τQGuj,=12∑i=1,j=1nτQui+τQujdQui,uj+12∑i=1,j=1mτQwi+τQwjdQwi,wj,+12∑i=1n∑j=1mτQui+τQwjdQui,wj.⩽12∑i=1,j=1nδGui+τGui+r+δGuj+τGuj+rdGui,uj+1+12∑i=1,j=1mτGxi+τGyi+s+τGxj+τGyj+sdGei,ej+1+12∑i=1n∑j=1mδGui+τGui+r+τGxj+τGyj+sdGui,ej+1.=12∑i=1,j=1nδGui+δGujdGui,uj+12∑i=1,j=1nτGui+τGujdGui,uj+r∑i=1,j=1ndGui,uj+12∑i=1,j=1nδGui+δGuj+12∑i=1,j=1nτGui+τGuj+r∑i=1,j=1n+12∑i=1,j=1mτGxi+τGyi+τGxj+τGyjdGei,ej+1+s∑i,j=1mdGei,ej+1+12∑i=1n∑j=1mδGui+τGujdGui,ej+1+r+s2∑i=1n∑j=1mdGui,ej+1+12∑i=1n∑j=1mτGxj+τGyjdGui,ej+1.=DDG+CDG+2rWG+2mn+n∑i=1nτGui+rn2+2sWeG+12∑i=1,j=1mτGxi+τGyi+τGxj+τGyjdGei,ej+1+12∑i=1n∑j=1mδGui+τGuidGui,ej+1+r+s2∑i=1n∑j=1mdGui,ej+1+12∑i=1n∑j=1mτGxj+τGyjdGei,ej+1.
IfG is C3,C4− free graph, then connection distance index (CD) of QG can also be determined by taking r=0 and s=0 in Theorem 3.Corollary 2.
IfG is a C3,C4− free graph, then(22)CDQG=DDG+CDG+2mn+n∑i=1nτGui+12∑i=1,j=1mτGxi+τGyi+τGxj+τGyjdGei,ej+1+12∑i=1n∑j=1mδGui+τGuidGui,ej+1+12∑i=1n∑j=1mτGxj+τGyjdGei,ej+1.Theorem 4.
IfT=TG is an total graph of G, then(23)CDTG=2CDG+2rWG+2sWeG+∑i=1n∑j=1mτGuidGui,ej+1+12∑i=1,j=1mτGxi+τGyi+τGxj+τGyjdGei,ej+1+r+s2∑i=1n∑j=1mdGui,ej+1+12∑i=1n∑j=1mτGxj+τGyjdGei,ej+1.Proof 8.
To avoid confusion, we take edgeei=xiyi=ujuk for some j and k. By lemma 4, τTGwi⩽τGxi+τGyi+s and τTGui=2τGui+r
Also(24)dTu,v=dGu,v,dTwi,wj=dGei,ej+1,dTui,wj=dGui,ei+1,CDTG=∑u,v⊆VQdTGu,vτTGu+τTGv,=12∑i=1,j=1nτTui+τQujdTui,uj+12∑i=1,j=1mτTwi+τTwjdTwi,wj+12∑i=1n∑j=1mτTui+τTwjdTui,wj⩽12∑i=1,j=1n2τGui+r+2τGuj+rdGui,uj+12∑i,j=1mτGxi+τGyi+s+τGxj+τGyj+sdGei,ej+1+12∑i=1n∑j=1m2τGui+r+τGxj+τGyj+sdGui,ej+1=∑i=1,j=1nτGui+τGujdGui,uj+r∑i=1,j=1ndGui,uj+12∑i=1,j=1nτGxj+τGyj+τGxj+τGyjdGei,ej+1+s∑i,j=1mdGei,ej+1+∑i=1n∑j=1mτGujdGui,ej+1+r+s2∑i=1n∑j=1mdGui,ej+1+12∑i=1n∑j=1mτGxj+τGyjdGui,ej+1=2CDG+2rWG+2sWeG+∑i=1n∑j=1mτGuidGui,ej+1+12∑i=1,j=1mτGxi+τGyi+τGxj+τGyjdGei,ej+1+r+s2∑i=1n∑j=1mdGui,ej+1+12∑i=1n∑j=1mτGxj+τGyjdGei,ej+1.
IfG is C3,C4− free graph, then connection distance index (CD) of TG can also be determined by taking r=0 and s=0 in Theorem 4.Corollary 3.
IfG is a C3,C4− free graph, then(25)CDTG=2CDG+12∑i=1,j=1mτGxi+τGyi+τGxj+τGyjdGei,ej+1+∑i=1n∑j=1mτGuidGui,ej+1+12∑i=1n∑j=1mτGxj+τGyjdGei,ej+1.


## 6. Conclusion

In previous section, first of all, connection numbers of old (black) vertices and new (white) were determined for derived graphFG where F∈S,R,Q,T (by Lemma 1, Lemma 2,Lemma 3 and Lemma 4 ) . It was found that exact values of connection numbers for C3,C4− free graph and SG were determined. Bounded values of RG, QG and TG for C3,C4− graphs were determined. Then, finally, exact and bounded values of connection distance (CD) of derived graphs SG,RG,QG,TG were determined by Theorems and Corollaries. International Academy of Mathematical Chemistry declared better reports for the chemical capability of the Zagreb connection indices than the ordinary Zagreb indices for the entropy and acentric factor of the octane isomers.



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*Source: 1439177-2022-07-18.xml*