# Computing Exact Values for Gutman Indices of Sum Graphs under Cartesian Product

Authors: Abdulaziz Mohammed Alanazi; Faiz Farid; Muhammad Javaid; Augustine Munagi
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Gutman index of a connected graph is a degree-distance-based topological index. In extremal theory of graphs, there is great interest in computing such indices because of their importance in correlating the properties of several chemical compounds. In this paper, we compute the exact formulae of the Gutman indices for the four sum graphs (S-sum, R-sum, Q-sum, and T-sum) in the terms of various indices of their factor graphs, where sum graphs are obtained under the subdivision operations and Cartesian products of graphs. We also provide specific examples of our results and draw a comparison with previously known bounds for the four sum graphs.


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## Body

## 1. Introduction

Theory of topological indices (TIs) started when Wiener discovered a close correlation between boiling points of certain alkanes and sums of the distances among pairs of vertices. Later, this calculated number was named Wiener Index [1]. After 25 years, Gutman and Trinajstić discovered degree-based indices (first and second Zagreb indices) which they used to compute the total π-electron energy of conjugate molecules [2]. Following these discoveries, many scientists began to introduce various TIs as invariant numbers for the prediction of the certain properties of molecular structures such as boiling point, freezing point, volume, density, vaporization, and weight. Two deeper approaches, namely, quantitative structures property relationships (QSPR) and quantitative structure activity relationships (QSAR) have also been used under the subject of cheminformatics (combination of chemistry, mathematics, statistics, and information sciences) and in conjunction with TIs, to find correlation values between the physical structures and chemical properties of molecules, see [3–5].TIs have been classified into three main classes depending upon degrees of nodes (vertices), distances among the vertices, and enumerative polynomials of the molecular graphs. Distance-based TIs are generally considered more important than the others. Some of the distance-based TIs are Wiener index [1], average distance index [6], Harary index [7, 8], degree distance index, and the Gutman index [9]. For more details, see [10–14].In graph theory, various operations such as union, intersection, addition, and Cartesian product are used to obtain the new graphs. Yan et al. [15] defined four subdivision-related operations S, R, Q, and T. They applied these operations on a connected graph G to obtain the four new graphs SG (subdivided graph), RG (triangle parallel graph), QG (line superposition graph), and TG (total graph), respectively. Afterwards, Das and Gutman [10] introduced the F-sum graphs using the operation of Cartesian product on the graphs FG1 and G2, where F∈S,R,QandT. Various hexagonal chains were later derived from these F-sum graphs, which have been found isomorphic to many chemical structures. They also determined the Wiener indices of the following S-sum G1+Gs2, R-sum G1+GR2, Q-sum G1+GQ2, and T-sum G1+GT2 graphs.Recently, Liu et al. [16] computed the first general Zagreb indices of the F-sum graphs. Akhter and Imran [17] found out the sharp bounds of the general sum-connectivity index for F-sum graphs. Ahmad et al. [18] discovered the exact formulae of the general sum-connectivity index for F-sum graphs, by improving the bounds. An et al. [19] determined the upper bounds of the degree distance indices for all the F-sum graphs. Pattabiraman and Bhat [20] derived the upper bounds of the Gutman index for all the F-sum graphs. For more studies, we refer to [21–32].In this paper, we obtain the exact values of the Gutman index for the F-sumG1+Gs2, G1+GR2, G1+GQ2, and G1+GT2 graphs. Moreover, the results are illustrated with special classes of the F-sum graphs and a comparison is drawn between the obtained exact values and the previously known bounded values.The sections of paper are organized as follows. Section2 comprises of preliminaries (some important definitions and statements of related lemmas). Section 3 contains the main result consisting of statements and proofs of theorems about Gutman indices of F-sum graphs. Lastly, Section 4 covers the applications of the main result to the computation of Gutman indices of particular classes of the F-sum graphs and a comparison among exact and known bounded values.


## 2. Preliminaries

We give a detailed consideration of two simple graphsG1 and G2. The degree of a vertex xdegxor dx is equal to the number of vertices connected to it. For each x,y∈VG1×G2, degree of the vertex x,y is denoted by degx,y. The distance dx,y between two vertices x,y∈VG is defined as the length of the shortest path between both the vertices x and y. Further details can be found in [33, 34].Definition 1.
(see [1]). The Wiener index WG of a connected graph G is defined as(1)WG=12∑x,y∈VGdx,y.Definition 2.
(see [11]). The degree distance index DDG of a connected graph G is defined as(2)DDG=12∑x,y∈VGdx,ydegx+degy.Definition 3.
(see [35]). The Gutman index GMG of a connected graph G is defined as(3)GMG=12∑x,y∈VGdx,ydegxdegy.
Yan et al. [15] defined four special graphs derivable from a given graph G by applying the four respective operations S, R, Q, and T on the graph G as follows:(i)
Subdivided graphSG is formed from G if distance of one between two adjacent pair of vertices is increased by two after inserting a new vertex between them. The vertices of G are named as black vertices, while new vertices are called white vertices.(ii)
Triangle parallel graphRG is formed from G if the new vertex corresponded to each edge of G is joined with the end vertices of the each respective edge. The vertices of G are named as black vertices, while new vertices are called white vertices.(iii)
Line superposition graphQG is formed when two white vertices obtained from SG are further joined if incident edges of these white vertices have one common end vertex in G.(iv)
Total graphTG is formed from RG by applying the further operation Q on it.
Figure1 illustrates the graphs obtained by the operations S, R, Q, and T based on the path graph P6.Figure 1
(a)P6, (b) SP6, (c) RP6, (d) QP6, and (e) TP6.
(a)(b)(c)(d)(e)Definition 4.
LetF∈S,R,QandT. Then, G1+GF2 is called a F-sum graph with vertex set VG1+GF2=VG1∪EG1×VG2 and u,v,x,y∈VG1+GF2 are adjacent such that either u=x and v,y∈EG2 or v=y and u,x∈EFG1.
Figure2 shows instances of S-sum G1+Gs2, R-sum G1+GR2, Q-sum G1+GQ2, and T-sum G1+GT2 graphs. We now state some important lemmas which are frequently used in the main results.Figure 2
(a)P4+Ps2. (b) P4+PR2. (c) P4+PQ2. (d) P4+PT2.
(a)(b)(c)(d)Lemma 1.
(see [36]). Let G1 and G2 be two simple and connected graphs.(a)
ForF∈S,R,Q,T, if both vertices x,y and w,z are black, then dx,y,w,z|G1+GF2=dx,w|FG1+dy,z|G2(b)
If one vertexx,y is white and second w,z is black with F∈S,R,Q,T, then dx,y,w,z|G1+GF2=dx,w|FG1+dy,z|G2Lemma 2.
(see [36]). Let G1 and G2 be two simple and connected graphs. If both vertices x,y and w,z are white vertices and F=S or R, then (4)dx,y,w,z|G1+GF2=2+dy,z|G2,Ifx=w,dx,w|FG1+dy,z|G2,Ifx≠w.Lemma 3.
(see [36]). Let G1 and G2 be two simple and connected graphs. If both vertices x,y and w,z are white vertices and F=Q or T, then(5)dx,y,w,z|G1+GF2=2+dy,z|G2,Ifx=w,1+dx,w|FG1+dy,z|G2,Ifx≠w,y≠z.


## 3. Main Results

This section is devoted to providing main theorems on the Gutman index of F-sum graphsG1+Gs2, G1+GR2, G1+GQ2, and G1+GF2. Consider the set VG1=u1,u2,u3,…,un of black vertices with EG1=k so that W=w1,w2,w3,…,wk consists of white vertices and VG2=v1,v2,v3,…,vm with EG2=l. Then, VG1+GF2=ui,vj:1≤i≤n,1≤j≤m∪wi,vj:1≤i≤k,1≤j≤m, where F∈S,R,Q,T.Theorem 1.
LetG1 and G2 be connected and simple graphs. If G1+Gs2 is the S-sum graph of G1 and G2, then(6)GMG1+Gs2=m2GMSG1+n2GMG2+4nkDDG2+16k2WG2+4km2−m+lm∑i,j=1ndui,uj|SG1dui+duj+2l2∑i,j=1ndui,uj|SG1+4lm∑r=1k∑i=1ndwr,ui|SG1.Proof.
Case 1: when both vertices are black,(7)A=12∑dui,vp,uj,vqdegui,vpdeguj,vq|G1+Gs2,fori,j=1 to n and p,q=1to m. For S-sum degui,vp=dui+dvp,(8)A=12∑i,j=1n∑p,q=1mdui,uj|SG1+dvp,vq|G2dui+dvpduj+dvq=12∑i,j=1n∑p,q=1mdui,uj|SG1+dvp,vq|G2duiduj+duidvq+dvpduj+dvpdvq=12∑i,j=1n∑p,q=1mdui,uj|SG1duiduj+duidvq+dvpduj+dvpdvq+12∑i,j=1n∑p,q=1mdvp,vq|G2duiduj+duidvq+dvpduj+dvpdvq=12∑i,j=1n∑p,q=1mdui,uj|SG1duiduj+12∑i,j=1n∑p,q=1mdui,uj|SG1duidvq+12∑i,j=1n∑p,q=1mdui,uj|SG1dvpduj+12∑i,j=1n∑p,q=1mdui,uj|SG1dvpdvq+12∑i,j=1n∑p,q=1mdvp,vq|G2duiduj+12∑i,j=1n∑p,q=1mdvpdvq|G2duidvq+12∑i,j=1n∑p,q=1mdvp,vq|G2dvpduj+12∑i,j=1n∑p,q=1mdvpdvq|G2dvpdvq.Substituting∑p,q=1mdvq=2lm,∑i,j=1nduj=2kn,(9)=m22∑i,j=1ndui,uj|SG1duiduj+lm∑i,j=1ndui,uj|SG1dui+lm∑i,j=1ndui,uj|SG1duj+12∑i,j=1ndui,uj|SG1∑p,q=1mdvpdvq=m22∑i,j=1ndui,uj|SG1duiduj+lm∑i,j=1ndui,uj|SG1dui+12∑p,q=1mdvp,vq|G2∑i,j=1nduiduj+kn∑p,q=1mdvp,vq|G2dvq+kn∑p,q=1mdvp,vq|G2dvp+n2GMG2.Substituting∑i,j=1nduiduj=4k2and∑p,q=1mdvpdvq=4l2,(10)=m22∑i,j=1ndui,uj|SG1duiduj+lm∑i,j=1ndui,uj|SG1dui+duj+2l2∑i,j=1ndui,uj|SG1+4k2WG2+2knDDG2+n2GMG2Case 2: when one vertex is white and the other is black,(11)B1=12∑dwr,vp,ui,vqdegwr,vpdegui,vq|G1+Gs2.Forr=1 to k, i=1 to n, and p,q=1 to m and degwr,vp=2,(12)B1=12∑r=1k∑i=1n∑p,q=1mdwr,ui|SG1+dvp,vq|G22dui+dvq=∑r=1k∑i=1n∑p,q=1mdwr,ui|SG1dui+∑r=1k∑i=1n∑p,q=1mdwr,ui|SG1dvq+∑r=1k∑i=1n∑p,q=1mdvp,vq|G2dui+∑r=1k∑i=1n∑p,q=1mdvp,vq|G2dvq=m2∑r=1k∑i=1ndwr,ui|SG1dui+2lm∑r=1k∑i=1ndwr,ui|SG1+4k2WG2+knDDG2.The summation of the distances between vertices with different colours is twice theB1, i.e.,(13)B=2m2∑r=1k∑i=1ndwr,ui|SG1dui+4lm∑r=1k∑i=1ndwr,ui|SG1+8k2WG2+2knDDG2.Case 3: when both vertices are white,(14)C=12∑dwr,vp,ws,vqdegwr,vpdegws,vq|G1+Gs2.This summation consists of two partsC=C1+C2 and degwr,vp=2, where(15)C1=12∑dwr,vp,ws,vqdegwr,vpdegws,vq|G1+Gs2:r=s,p≠q,C2=12∑dwr,vp,ws,vqdegwr,vpdegws,vq|G1+Gs2:r≠s,C1=12∑dwr,vp,ws,vqdegwr,vpdegws,vq|G1+Gs2=12∑2+dvp,vq22|G1+Gs2=4∑r,s=1,r=sk∑p,q=1,p≠qm+2∑r,s=1,r=sk∑p,q=1,p≠qmdvp,vq|G2=4km2−m+4kWG2,C2=12∑dwr,vp,ws,vqdegwr,vpdegws,vq|G1+Gs2,r≠s=12∑p=1m∑q=1m∑r,s=1,r≠skdwr,ws|SG14+12∑p=1m∑p=1m∑r,s=1,r≠skdvp,vq|G24=2m2∑r,s=1,r≠skdwr,wsSG1+4k2−kWG2,C=4km2−m+4kWG2+2m2∑r,s=1,r≠skdwr,ws|SG1+4k2−kWG2=4km2−m+2m2∑r,s=1,r≠skdwr,ws|SG1+4k2WG2.
Now, Gutman index ofG1+sG2 is given by(16)GMG1+Gs2=A+B+C,i.e.,GMG1+Gs2=m22∑i,j=1ndui,uj|SG1duiduj+lm∑i,j=1ndui,uj|SG1dui+duj+2l2∑i,j=1ndui,uj|SG1+4k2WG2+2knDDG2+n2GMG2+2m2∑r=1k∑i=1ndwr,ui|SG1dui+4lm∑r=1k∑i=1ndwr,ui|SG1+8k2WG2+2knDDG2+4km2−m+2m2∑r,s=1,r≠skdwr,ws|SG1+4k2WG2,GMG1+Gs2=m212∑i,j=1ndui,uj|SG1duiduj+∑r=1k∑i=1ndwr,uiSG12dui+2∑r,s=1,r≠skdwr,ws|SG1+n2GMG2+4knDDG2+16k2WG2+4km2−m+lm∑i,j=1ndui,uj|SG1dui+duj+2l2∑i,j=1ndui,uj|SG1+4lm∑r=1k∑i=1ndwr,ui|SG1,GMG1+Gs2=m2GMSG1+n2GMG2+4nkDDG2+16k2WG2+4km2−m+lm∑i,j=1ndui,uj|SG1dui+duj+2l2∑i,j=1ndui,uj|SG1+4lm∑r=1k∑i=1ndwr,ui|SG1.Theorem 2.
LetG1 and G2 are connected and simple graphs. If G1+GR2 is the R-sum graph of G1 and G2, then(17)GMG1+GR2=m2GMRG1+n2GMG2+6nkDDG2+36k2WG2+4km2−m+2lm∑i,j=1ndui,uj|RG1dui+duj+2l2∑i,j=1ndui,uj|RG1+4lm∑r=1k∑i=1ndwr,ui|RG1.Proof.
1) When both vertices are black,(18)A=12∑dui,vp,uj,vqdegui,vpdeguj,vq|G1+GR2,fori,j=1 to n and p,q=1to m. For R-sum degui,vp=2dui+dvp,(19)A=12∑i,j=1n∑p,q=1mdui,uj|RG1+dvp,vq|G22dui+dvp2duj+dvq=12∑i,j=1n∑p,q=1mdui,uj|RG1+dvp,vq|G24duiduj+2duidvq+2dvpduj+dvpdvq=12∑i,j=1n∑p,q=1mdui,uj|RG14duiduj+2duidvq+2dvpduj+dvpdvq+12∑i,j=1n∑p,q=1mdvp,vq|G24duiduj+2duidvq+2dvpduj+dvpdvq=2∑i,j=1n∑p,q=1mdui,uj|RG1duiduj+∑i,j=1n∑p,q=1mdui,uj|RG1duidvq+∑i,j=1n∑p,q=1mdui,uj|RG1dvpduj+12∑i,j=1n∑p,q=1mdui,uj|RG1dvpdvq+2∑i,j=1n∑p,q=1mdvp,vq|G2duiduj+∑i,j=1n∑p,q=1mdvp,vq|G2duidvq+∑i,j=1n∑p,q=1mdvp,vq|G2dvpduj+12∑i,j=1n∑p,q=1mdvp,vq|G2dvpdvq.Substituting∑p,q=1mdvq=2lmand∑i,j=1nduj=2kn,(20)=2m2∑i,j=1ndui,uj|RG1duiduj+2lm∑i,j=1ndui,uj|RG1dui+2lm∑i,j=1ndui,uj|RG1dui+12∑i,j=1ndui,uj|RG1∑p,q=1mdvpdvq+2∑p,q=1mdvp,vq|G2∑i,j=1nduiduj+2kn∑p,q=1mdvp,vq|G2dvq+2kn∑p,q=1mdvp,vq|G2dvp+n2GMG2.Substituting∑i,j=1nduiduj=4k2and∑p,q=1mdvpdvq=4l2,(21)A=2m2∑i,j=1ndui,uj|RG1duiduj+2lm∑i,j=1ndui,uj|RG1dui+duj+2l2∑i,j=1ndui,uj|SG1+16k2WG2+4knDDG2+n2GMG2.Case 2: when one vertex is white and the other is black,(22)B1=12∑dwr,vp,ui,vqdegwr,vpdegui,vq|G1+GR2.Forr=1 to k, i=1 to n, and p,q=1 to m and degwr,vp=2,(23)B1=12∑r=1k∑i=1n∑p,q=1mdwr,ui|RG1+dvp,vq|G222dui+dvq=2∑r=1k∑i=1n∑p,q=1mdwr,ui|RG1dui+∑r=1k∑i=1n∑p,q=1mdwr,ui|RG1dvq+2∑r=1k∑i=1n∑p,q=1mdvp,vq|G2dui+∑r=1k∑i=1n∑p,q=1mdvp,vq|G2dvq=2m2∑r=1k∑i=1ndwr,ui|RG1dui+2lm∑r=1k∑i=1ndwr,ui|RG1+8k2WG2+knDDG2.The summation of the distances between vertices with different colours is twice theB1, i.e.,(24)B=4m2∑r=1k∑i=1ndwr,ui|RG1dui+4lm∑r=1k∑i=1ndwr,ui|RG1+16k2WG2+2knDDG2.Case 3: when both vertices are white, C can be calculated similar to that in the previous theorem:(25)C=4km2−m+2m2∑r,s=1,r≠skdwr,ws|RG1+4k2WG2.
Now, Gutman index ofG1+GR2 is given by(26)GMG1+GR2=A+B+C,i.e.,GMG1+GR2=2m2∑i,j=1ndui,uj|RG1duiduj+2lm∑i,j=1ndui,uj|RG1duiduj+2l2∑i,j=1ndui,uj|RG1+16k2WG2+4knDDG2+n2GMG2+4m2∑r=1k∑i=1ndwr,ui|RG1dui+4lm∑r=1k∑i=1ndwr,ui|RG1+16k2WG2+2knDDG2+4km2−m+2m2∑r,s=1,r≠skdwr,ws|RG1+4k2WG2,GMG1+GR2=m22∑i,j=1ndui,uj|RG1duiduj+∑r=1k∑i=1ndwr,ui|RG14dui+2∑r,s=1,r≠skdwr,ws|RG1+n2GMG2+6knDDG2+36k2WG2+4km2−m+2lm∑i,j=1ndui,uj|RG1dui+duj+2l2∑i,j=1ndui,uj|RG1+4lm∑r=1k∑i=1ndwr,ui|RG1,GMG1+GR2=m2GMRG1+n2GMG2+6nkDDG2+36k2WG2+4km2−m+2lm∑i,j=1ndui,uj|RG1dui+duj+2l2∑i,j=1ndui,uj|RG1+4lm∑r=1k∑i=1ndwr,ui|RG1,Theorem 3.
LetG1 and G2 be two simple and connected graphs. If G1+GQ2 is the Q-sum graph of G1 and G2, then(27)GMG1+GQ2=m2GMQG1+n2GMG2+n6k−tDDG2+6k−t2WG2+12m2−m4k−t2+16k−7t+lm∑i,j=1ndui,uj|QG1dui+duJ+2l2∑i,j=1ndui,uj|QG1+2lm∑r=1k∑i=1ndwr,ui|QG1degwr,vp.Proof.
Case 1: when both vertices are black, A can be determined similar to that in Case 1 of Theorem1:(28)A=m22∑i,j=1ndui,uj|QG1duiduj+lm∑i,j=1ndui,uj|QG1duiduj+2l2∑i,j=1ndui,uj|QG1+4k2WG2+2knDDG2+n2GMG2.Case 2: when one vertex is white and the other is black,(29)B1=12∑dwr,vp,ui,vqdegwr,vpdegui,vq|G1+GQ2.Forr=1 to k, i=1 to n and p,q=1 to m,(30)B1=12∑r=1k∑i=1n∑p,q=1mdwr,ui|SG1+dvp,vq|G2degwr,vpdui+dvq=12∑r=1k∑i=1n∑p,q=1mdwr,ui|QG1degwr,vpdui+12∑r=1k∑i=1n∑p,q=1mdwr,ui|QG1degwr,vpdvq+12∑r=1k∑i=1n∑p,q=1mdvp,vq|G2degwr,vpdui+12∑r=1k∑i=1n∑p,q=1mdvp,vq|G2degwr,vpdvq=m22∑r=1k∑i=1ndwr,ui|QG1degwr,vpdui+lm∑r=1k∑i=1ndwr,ui|QG1degwr,vp+2k4k−tWG2+12n4k−tDDG2.The summation of the distances between vertices with different colours is twice theB1, i.e.,(31)B=m2∑r=1k∑i=1ndwr,ui|QG1degwr,vpdui+2lm∑r=1k∑i=1ndwr,ui|QG1degwr,vp+4k4k−tWG2+n4k−tDDG2.Case 3: When both vertices are white,(32)C=12∑dwr,vp,ws,vqdegwr,vpdegws,vq|G1+GQ2.This summation consists of three partsC=C1+C2+C3, where(33)C1=12∑dwr,vp,ws,vqdegwr,vpdegws,vq|G1+GQ2:r=s,p≠q,C2=12∑dwr,vp,ws,vqdegwr,vpdegws,vq|G1+GQ2:r≠s,p=q,C3=12∑dwr,vp,ws,vqdegwr,vpdegws,vq|G1+GQ2:r≠s,p≠q,C1=12∑2+dvp,vqdegwr,vpdegws,vq|G1+GQ2:r=s,p≠q,C1=∑r,s=1,r=sk∑p,q=1,p≠qmdegwr,vpdegws,vq+12∑r,s=1,r=sk∑p,q=1,p≠qmdvp,vq|G2degwr,vpdegws,vq,C1=m2−m16k−t+9t+WG216k−t+9t,C2=12∑p,q=1,p=qm∑r,s=1,r≠skdwr,ws|QG1degwr,vpdegws,vq=m2∑r,s=1,r≠skdwr,ws|QG1degwr,vpdegws,vq,C3=12∑r,s=1,r≠sk∑p,q=1,p≠qm1+dwr,ws|QG1+dvp,vq|G2degwr,vpdegws,vq=12∑r,s=1,r≠sk∑p,q=1,p≠qmdegwr,vpdegws,vq+12∑r,s=1,r≠sk∑p,q=1,p≠qmdwr,ws|QG1degwr,vpdegws,vq+12∑r,s=1,r≠sk∑p,q=1,p≠qmdvp,vq|G2degwr,vpdegws,vq=12m2−m4k−t2−16k−7t+12m2−m∑r,s=1,r≠skdwr,ws|QG1degwr,vpdegws,vq+4k−t2−16k−7tWG2,C=m2−m16k−7t+WG216k−7t+m2∑r,s=1,r≠skdwr,ws|QG1degwr,vpdegws,vq+12m2−m4k−t2−16k−7t+12m2−m∑r,s=1,r≠skdwr,ws|QG1degwr,vpdegws,vq,C=12m2−m4k−t2+16k−7t+4k−t2WG2+m22∑r,s=1,r≠skdwr,ws|QG1degwr,vpdegws,vq.
Now, Gutman index ofG1+GQ2 is given by(34)GMG1+GQ2=A+B+C,i.e.,using values of A, B, and C from Case 1, Case 2, and Case 3, respectively,(35)GMG1+GQ2=m22∑i,j=1ndui,uj|QG1duiduj+lm∑i,j=1ndui,uj|QG1dui+duj+2l2∑i,j=1ndui,uj|QG1+4k2WG2+2knDDG2+n2GMG2+m2∑r=1k∑i=1ndwr,ui|QG1degwr,vpdui+2lm∑r=1k∑i=1ndwr,ui|QG1degwr,vp+4k4k−tWG2+n4k−tDDG2+12m2−m4k−t2+16k−7t+4k−t2WG2+m22∑r,s=1,r≠skdwr,ws|QG1degwr,vpdegws,vq,GMG1+GQ2=m212∑i,j=1ndui,uj|QG1duiduj+∑r=1k∑i=1ndwr,ui|QG1degwr,vpdui+12∑r,s=1,r≠skdwr,ws|QG1degwr,vpdegws,vq+n2GMG2+n6k−tDDG2+WG24k2+4k4k−t+4k−t2+12m2−m4k−t2+16k−7t+lm∑i,j=1ndui,uj|QG1duiduj+2l2∑i,j=1ndui,uj|QG1+2lm∑r=1k∑i=1ndwr,ui|QG1degwr,vp,GMG1+GQ2=m2GMQG1+n2GMG2+n6k−tDDG2+6k−t2WG2+12m2−m4k−t2+16k−7t+lm∑i,j=1ndui,uj|QG1dui+duj+2l2∑i,j=1ndui,uj|QG1+2lm∑r=1k∑i=1ndwr,ui|QG1degwr,vp.Theorem 4.
LetG1 and G2 be two simple and connected graphs. If G1+GT2 is the T-sum graph of G1 and G2, then(36)GMG1+GT2=m2GMTG1+n2GMG2+n8k−tDDG2+8k−t2WG2+12m2−m4k−t2+16k−7t+4lm∑i,j=1ndui,uj|TG1dui+2l2∑i,j=1ndui,uj|TG1+2lm∑r=1k∑i=1ndwr,ui|TG1degwr,vp.Proof.
Case 1: when both vertices are black, A can be calculated similar to that in Case 1 of Theorem2:(37)A=2m2∑i,j=1ndui,uj|TG1duiduj+2lm∑i,j=1ndui,uj|TG1duiduj+2l2∑i,j=1ndui,uj|TG1+16k2WG2+4knDDG2+n2GMG2.Case 2: when one vertex is white and the other is black,(38)B1=12∑dwr,vp,ui,vqdegwr,vpdegui,vq|G1+GT2.Forr=1 to k, i=1 to n, and p,q=1 to m,(39)B1=12∑r=1k∑i=1n∑p,q=1mdwr,ui|TG1+dvp,vq|G2degwr,vp2dui+dvq=∑r=1k∑i=1n∑p,q=1mdwr,ui|TG1degwr,vpdui+12∑r=1k∑i=1n∑p,q=1mdwr,ui|TG1degwr,vpdvq+∑r=1k∑i=1n∑p,q=1mdvp,vq|G2degwr,vpdui+12∑r=1k∑i=1n∑p,q=1mdvp,vq|G2degwr,vpdvq=m2∑r=1k∑i=1ndwr,ui|TG1degwr,vpdui+lm∑r=1k∑i=1ndwr,ui|TG1degwr,vp+4k4k−tWG2+12n4k−tDDG2.The summation of the distances between vertices with different colours is twice theB1, i.e.,(40)B=2m2∑r=1k∑i=1ndwr,ui|TG1degwr,vpdui+2lm∑r=1k∑i=1ndwr,ui|TG1degwr,vp+8k4k−tWG2+n4k−tDDG2.Case 3: when both vertices are white, C can be calculated similar to that in Case 3 of Theorem3:(41)C=12m2−m4k−t2+16k−7t+4k−t2WG2+m22∑r,s=1,r≠skdwr,ws|QG1degwr,vpdegws,vq.Now, Gutman index ofG1+GT2 is given by(42)GMG1+GT2=A+B+C,i.e.,using values of A, B, and C from Case 1, Case 2, and Case 3, respectively,(43)GMG1+GT2=2m2∑i,j=1ndui,uj|TG1duiduj+2lm∑i,j=1ndui,uj|TG1duiduj+2l2∑i,j=1ndui,uj|TG1+16k2WG2+4knDDG2+n2GMG2+2m2∑r=1k∑i=1ndwr,ui|TG1degwr,vpdui+2lm∑r=1k∑i=1ndwr,ui|TG1degwr,vp+8k4k−tWG2+n4k−tDDG2+12m2−m4k−t2+16k−7t+4k−t2WG2+m22∑r,s=1,r≠skdwr,ws|TG1degwr,vpdegws,vq,GMG1+GT2=m22∑i,j=1ndui,uj|TG1duiduj+2∑r=1k∑i=1ndwr,ui|TG1degwr,vpdui+12∑r,s=1,r≠skdwr,ws|TG1degwr,vpdegws,vq+n2GMG2+n8k−tDDG2+WG216k2+8k4k−t+4k−t2+12m2−m4k−t2+16k−7t+4lm∑i,j=1ndui,uj|TG1dui+2l2∑i,j=1ndui,uj|TG1+2lm∑r=1k∑i=1ndwr,ui|TG1degwr,vp,GMG1+GT2=m2GMTG1+n2GMG2+n8k−tDDG2+8k−t2WG2+12m2−m4k−t2+16k−7t+4lm∑i,j=1ndui,uj|TG1dui+2l2∑i,j=1ndui,uj|TG1+2lm∑r=1k∑i=1ndwr,ui|TG1degwr,vp.


## 4. Discussion and Conclusion

In this section, we apply the main results of Section3 by taking G1 equal to, firstly, Pn, and secondly, Cn, while G2 is set equal to Pm. The Wiener index degree, distance index, and Gutman index of Pn are WPn=nn2−1/6, DDPn=nn−12n−1/3, and GMPn=n−12n2−4n+3/3, respectively. Now, we construct Tables 1 and 2.Table 1
Wiener index, degree distance, and Gutman Index ofFPn.
FWFPnDDFPnGMFPnS2nn−12n−1/32n−12n−14n−3/32n−18n2−16n+9/3Rn−12n2+2n−3/34n+1n−126nn−12Q2nn−1n+1/34n3−6n2+2n−223n3−9n2+10n−5Tnn−14n+1/62/38n3−15n2+10n−632n3−96n2+115n−66/3Table 2
∑i,j=1ndui,uj|FPn, ∑r=1k∑i=1ndwr,ui|FPn, and ∑i,j=1Ndui,uj|FPndui+duj.
F∑i,j=1ndui,uj|FPn∑r=1k∑i=1ndwr,ui|FPn∑i,j=1Ndui,uj|FPndui+dujS2nn−1n+1/3nn−12n−1/34nn−12n−1/3Rnn2−1/3nn2−1/32nn−12n−1/3Qnn−1n+4/3nn−1n+1/322n3+3n2−11n+6/3Tnn−1n+1/3nn−1n+1/32nn−12n−1/3Then, the following results are found by using Theorems1–4 and Tables 3 and 4:(44)GMPn+PSm=2m2n−18n2−16n+93+n2m−12m2−4m+33+4nn−1mm−12m−13+4mnn−1m−12m−13+8n−12mm2−13+4mn−1m−1+4mnm−1n−12n−13+8m−12nn2−16+4mm−nn−12n−13,GMPn3+PRm=6m2nn2−1+n2m−12m2−4m+33+2nmn−1m−12m−1+6n−12mm2−1+4mn−1m−1+4mnm−1n−12n−13+2m−12nn2−13+4mnm−1n2−13,GMPn+PQm=2m23n3−9n2+10n−5+n2m−12m2−4m+33+mn6n−8m−12m−13+6n−82mm2−16+m2−m4n−62+16n−1−142+2mm−12n3+3n2−11n+63+2m−12nn+4n−13+2mm−14n3−3n2−n−63,GMPn+PTm=m232n3−96n2+115n−663+n2m−12m2−4m+33+mn8n−10m−12m−13+8n−102mm2−16+m2−m4n−62+16n−1−142+4mnm−1n−12n−13+2m−12nn2−13+2mm−14n3−3n2−n−63.Table 3
Wiener index and degree distance ofFCn.
FWFCnDDFCnGMFCnSn34n34n3Rnn2+2n−1/2nn+23n−1n9n2+12n−4/2ifnis evenn9n2+12n−5/2ifnis oddQnn2+2n−1/2nn+23n−1n9n2+12n−4/2ifnis evenn9n2+12n−5/2ifnis oddTn2n+1/24n2n+18n2n+1Table 4
∑i,j=1ndui,uj|FPn and ∑r=1k∑i=1ndwr,ui|FPn of FCn.
F∑i,j=1ndui,uj|FCn∑r=1k∑i=1ndwr,ui|FCnSn3/2ifnis evennn2−1/2ifnis oddn3/2ifnis evennn2+1/2ifnis oddRn3/4ifnis evennn2−1/4ifnis oddn2n+2/4ifnis evennn+12/4ifnis oddQnn2+4n−4/4ifnis evennn2+4n−5/4ifnis oddn2n+2/4ifnis evennn+12/4ifnis oddTn3/4ifnis evennn2−1/4ifnis oddn2n+2/4ifnis evennn+12/4ifnis oddIfCn is cycle of n vertices, then the Wiener index, degree distance index, and Gutman index of cycle are as follows:(45)WCn=n38,ifnis even,nn2−18,ifnis odd,DDCn=GMCn=n32,ifnis even,nn2−12,ifnis odd.It is very essential to know that, in cycle,dui=2 and(46)degwr,vp=2ifF=SorR,4ifF=QorT.Then, the following results are found by using Theorems1–4 and Tables 3 and 4:(47)GMCn+PSm=n2m−12m2−4m+33+4n2mm−12m−13+8n2mm2−13+4mn−1m−1+4m2n3+m−1n35m−1,ifnis even,n35m−1−nm−1ifnis odd,GMCn+PRm=n2m−12m2−4m+33+2n2mm−12m−1+6n2mm2−1+4mn−1m−1+m2n9n2+12n−42+m−1n2n7m−12+2m+0ifnis even,−n2m−122ifnis odd,GMCn+PQn=n2m−12m2−4m+33+2n2mm−12m−1+6n2mm2−1+8nn+1m2−m+m2n9n2+12n−42+nm−1n27m−12+2n5m−1−6m+2+0ifnis even,−n2ifnis odd,GMCn+PTm=n2m−12m2−4m+33+8n2mm−12m−13+32n2mm2−13++8nn+1m2−m+8m2n2n+1+nm−1n29m−12+4mn+0ifnis even,−nm−122ifnis odd.After deriving formulae, a comparison among the exact values, computed values, and bounded values is also drawn in Table5.Table 5
Comparision between exact, computed, and bounded values.
No.F-sum graphExact valueComputed valueBounded value1GMP3+PS250550510242GMP3+PR268968920023GMP3+PQ261961913934GMP3+PT290790724735GMC3+PS292192128326GMC3+PR21341134136147GMC3+PQ21617161740528GMC3+PT2237323736032Now, we close our discussion with the comments that the upper bounds for the Gutman indices on the F-sum (S-sum, R-sum, Q-sum, and T-sum) graphs are obtained in [20]. In this paper, we considered for the improvement of the already existing bounds and determined the exact values of the Gutman indices for the F-sum graphs. However, the problem is still open to find the exact values of other distance-based TIs for these sum graphs.



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