# On the Exact Values of HZ-Index for the Graphs under Operations

Authors: Dalal Awadh Alrowaili; Saira Javed; Muhammad Javaid
Journal: Journal of Mathematics
 (2021)

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## Abstract

Topological index (TI) is a function from the set of graphs to the set of real numbers that associates a unique real number to each graph, and two graphs necessarily have the same value of the TI if these are structurally isomorphic. In this note, we compute theHZ−index of the four generalized sum graphs in the form of the various Zagreb indices of their factor graphs. These graphs are obtained by the strong product of the graphs G and DkG, where Dk∈Sk,Rk,Qk,Tk represents the four generalized subdivision-related operations for the integral value of k≥1 and DkG is a graph that is obtained by applying Dk on G. At the end, as an illustration, we compute the HZ−index of the generalized sum graphs for exactly k=1 and compare the obtained results.


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## Body

## 1. Introduction

A structural formula of a chemical compound is represented by a molecular graph, where atoms and bonds between atoms are represented by the vertices and edges of the molecular graphs, respectively. A topological index (TI) is a mathematical tool which associates a real number to a graph under certain conditions. For two graphs, a TI remains constant if the graphs are isomorphic (see [1–3]). These are used to study different physical attributes, biological activities, and chemical reactivities such as viscosity, critical temperatures (boiling, freezing, melting, and flash points) [4, 5], vapor pressure, surface tension, stability, weight, density, solubility, and connectivity [6–8] in the field of chemical engineering, pharmaceutical industries, and drugs discoveries. TIs are also used in the subject of cheminformatics to study the quantitative structural activity and property relationships (see [9–11]).In 1947, the very first TI is introduced by Winer to check the critical temperature of paraffin [12]. Trinajstic and Gutman (1972) [13] defined the first and second Zagreb indices that are used to compute the different structure base characteristics of the molecular graphs. After that, many degree, distance, and polynomials based TIs came into existence but the degree-based indices got more attention of the researchers (see [14–16]). For various results on TIs of different graphs, see [17–20]. In 2008, Zhou and Trinajstić defined the general sum connectivity (GSC) index and discussed its various properties [21]. Shirdel et al. [22] studied the concept of hyper-Zagreb index (HZ−index) as a particular case of the GSC index. In addition, the results for the index HZ under the operation of Cartesian, composition, join, and disjunction of graphs can be found in [23–25].On the other hand, for the studies of the complex graphs, operations for graphs play a key role. Yan et al. (2007) defined four types of operations related to the subdivision ofG and computed the Wiener indices of the derived graphs D1G, where D1∈S1,R1,Q1,T1 [26]. Taeri et al. (2009) gave the construction of the D1-sum graphs GD1+H (Cartesian product of F1G and H) and computed their Wiener indices, where H and G are assumed to be two connected graphs [27]. Furthermore, Deng et al. [28], Akhter and Imran [29], Chu et al. [30], and Liu et al. [31] computed the various indices of these graphs with the help of the Cartesian product.Liu et al. (2019) [32] extended these operations for any integral value of k and obtained the generalized derived graphs DkG of the graph G, where Dk∈Sk,Rk,Qk,Tk. Moreover, using the concept of Cartesian product of graphs, they constructed the generalized sum graphs or Dk-sum graphs (denoted by GDk+H) and computed their first and second Zagreb indices.Javaid et al. (2021) [33] redefined these graphs using strong product and computed their Zagreb indices (first and second). In this development, we compute hyper-Zagreb indices (HZ−index) for these graphs in terms of various degree-based TIs of their factor graphs, where these generalized sum graphs are obtained with the help of strong product. The remaining paper is settled as follows. Section 2 contains the notations and key concepts which are utilized in methodology, Section 3 deals main results, and Section 4 covers examples and conclusion.


## 2. Preliminaries

This section explains the basic definitions and terminologies.Definition 1.
LetG=VG,EG be a (molecular) graph with VG and EG as sets of vertices and edges, respectively. The degree of a vertex v∈VG is the number of edges which are incident on v and denoted by dv.Definition 2 (see [13, 34]).
For a graphG, the first, second, and forgotten Zagreb indices are defined as follows: M1G=∑z∈VGd2z=∑zt∈EGdz+dt, M2G=∑zt∈EGdz×dt, and FG=∑z∈VGd3z=∑zt∈EGd2z+d2t.
These indices have been used to find the various properties of molecular graphs such as entropy,π-electron energy, and heat capacity. These are also used in the studies of the molecular structural relationships such as QSPR and QSAR [13, 35–37]. However, the hyper-Zagreb index of a graph G (given below) is studied by Shirdel et al. in 2013 [22]:(1)HZG=∑yz∈EGdy+dz2.Definition 3 (see [32]).
For some integral value ofk≥1, the graphs obtained by the generalized subdivision-related operations are defined as follows:(i)
SkG is a graph that is obtained by inserting k vertices in each edge of G(ii)
RkG is a graph obtained from SkG by joining the vertices which are adjacent in G(iii)
QkG is a graph obtained from SkG by joining the new vertices which are on the incident edges in G for each of its vertex(iv)
TkG is obtained from SkG after using both Rk and Qk, respectively
Fork=3, see Figure 1.Figure 1
(a)G1≅P4, (b) S3P4, (c) R3P4, (d) Q3P4 and (e) T3P4.
(a)(b)(c)(d)(e)Definition 4 (see [33]).
LetG1 and G2 be two graphs, Dk∈Sk,Rk,Qk,Tk be generalized subdivision-related operations, and DkG1 be a graph obtained using Dk on G1 having edge-set EDkG1 and vertex-set VDkG1. The generalized sum graph G1⊠DkG2 under the operation of strong product is a graph having vertex-set VG1⊠DkG2=VDkG1×VG2=VG1∪kEG1×VG2 such that two vertices r1,s1 and r2,s2 of VG1Dk⊠G2 are adjacent iff [r1=r2 in VG1 and s1 is adjacent to s2 in EG2] or [s1=s2 in VG2 and s1 is adjacent to s2 in EG1] or [r1 is adjacent to r2 in EDkG1 and s1 is adjacent to s2 in EG2], where k≥1 is a positive integer. For more explanation, see Figures 2 and 3.Figure 2
(a)P3⊠S2P3 and (b) P3⊠R2P3.
(a)(b)Figure 3
(a)P3⊠Q2P3 and (b) P3⊠T2P3.
(a)(b)


## 3. Main Results

The main developments are covered by this section.Theorem 1.
Fork≥1, the HZ-index of G1⊠SkG2 is(2)HZG1⊠SkG2=8eG1M1G2+nG1HZG2+4eG2M1G1+4eG1HZG2+M1G1HZG2+4M1G1M1G2+nG2HZS1G1+4eG2M1S1G1+2eG1M1G2+4eG2HZS1G2+2M1G2M1S1G1+M1G2HZS1G1+16k−1eG1nG2+M1G2+4eG2+HZ1G1FG2+2M1G2HZS1G1+2M1G2M1SG1+2FG2M1SG1+16eG1eG2+4eG1FG2+2eG2M1G1+4k−1eG18eG2+2HZG2+8M1G2.Proof.
Let the degree of a vertexr,s∈G1⊠SkG2 be denoted by dr,s:(3)HZG1⊠SkG2=∑r1,s1r2,s2∈EG1⊠SkG2dr1,s1+dr2,s22=∑rϵVG1∑s1s2ϵEG2dr,s1+dr,s22+∑r1r2ϵESkG1∑sϵVG2dr1,s+dr2,s2+∑r1r2ϵESkG1∑s1s2ϵVG2dr1,s1+dr2,s22=∑A+∑B+∑C.
Consider(4)∑A=∑rϵVG1∑s1s2ϵEG2dr,s1+dr,s22=∑rϵVG1∑s1s2ϵEG22dr+ds1+ds2+drds1+ds22=∑rϵVG1∑s1s2ϵEG24drds1+ds2+d2s1+ds2+2ds1ds2+4d2r+2drd2s1+d2s2+2ds1ds2+d2rd2s1+d2s2+2ds1ds2+4d2rds1+ds2=8eG1M1G2+nG1HZG2+4eG2M1G1+4eG1HZG2+M1G1HZG2+4M1G1M1G2,∑B=∑r1r2ϵESkG1∑sϵVG2dr1,s+dr2,s2=∑r1ϵVG1,r2ϵVSkG1−G1∑sϵVG2dr1,s+dr2,s2+∑r1,r2ϵVSkG1−G1∑sϵVG2dr1,s+dr2,s2=∑B1+∑B2,∑B1=∑r1r2ϵESkG1r1ϵVG1,r2ϵVSkG1−VG1∑sϵVG2dr1,s+dr2,s2=∑r1r2ϵESkG1r1ϵVG1,r2ϵVSkG1−VG1∑sϵVG2dr1+dr2+ds+dr1+dr2ds2=∑r1r2ϵESkG1r1ϵVG1,r2ϵVSkG1−VG1∑sϵVG2d2r1+d2r2+2dr1dr2+2dsdr1+dr2+d2s+2dsd2r1+d2r2+2dr1dr2+2d2sdr1+dr2+d2sd2r1+d2r2=nG2HZS1G1+4eG2M1S1G1+2eG1M1G2+4eG2HZS1G1+2M1G2M1S1G1+M1G2HZS1G1,∑B2=∑r1r2ϵESkG1r1,s2ϵVSkG1−VG1∑sϵVG2dr1,s+dr2,s2=∑r1r2ϵESkG1r1,r2ϵVSkG1−VG1∑sϵVG2dr1+dr1ds+dr2+dr2ds2=∑r1r2ϵESkG1r1,r2ϵVSkG1−VG1∑sϵVG24+4ds2=∑r1r2ϵESkG1r1,r2ϵVSkG1−VG1∑sϵVG216+16d2s+32ds.
Since in this caseESkG1=k−1EG1, we have(5)=∑sϵVG216k−1eG11+d2s+2ds=16k−1eG1nG2+M1G2+4eG2∑C=∑r1r2ϵESkG1dr1,s1+dr2,s22=∑r1r2ϵESkG1r1ϵVG1,r2ϵSkG1−VG1∑s1s2ϵVG2dr1,s1+dr2,s22+∑r1r2ϵESkG1r1,r2ϵVSkG1−VG1∑s1s2ϵVG2dr1,s1+dr2,s22=∑C1+∑C2,∑C1=∑r1r2ϵESkG1r1ϵVG1,r2ϵSkG1−VG1∑s1s2ϵVG2dr1,z1+dr2,s22=∑r1ϵVG1,r2ϵVSkG1−VG1∑s1s2ϵVG2dr1+dr2+ds1+dr1ds1+dr2ds22=HZG1FG2+2M1G2HZS1G1+2M1G2M1SG1+2FG2M1SG1+16eG1eG2+4eG1FG2+2eG2M1G1,∑C2=∑r1r2ϵESkG1r1,r2ϵVSkG1−VG1∑s1s2ϵVG2dr1,s1+dr2,s22=∑r1r2ϵESkG1r1,r2ϵVSkG1−VG1∑s1s2ϵVG24+2ds1+ds22=∑r1r2ϵESkG1r1,r2ϵVSkG1−VG1∑s1s2ϵVG216+4dz1+dz22+16dz1+dz2=4k−1eG18eG2+2HZG2+8M1G2.
Hence, we obtained our required result.Theorem 2.
Fork≥1, the HZ-index of G1⊠RkG2 is(6)HZG1⊠RkG2=8nG2+6eG2FG1+nG1+20eG1FG2+8FG1FG2+24eG2M1G1+36eG1M1G2+24M1G1M1G2+24FG1M1G2+8nG2eG1+8k−1eG1nG2+FG2+4eG2+3M1G2+48eG1eG2+12FG2M1G1+2M2G24eG1+nG1+knG2+6eG2+3M1G2+2M2G2×12∑vϵVG1dG14v−dG13v+∑vϵVG1rdG1udG1v+∑vϵVG1dG12v∑uϵVG1uvϵEG1dG1u−2M2G1+M1G15e2+5M1G2+5M2G2+kM3G1+2M2G16eG2+3M1G2+2M2G2+nG2+2eG1M1G2.Proof.
Let the degree of a vertexr,s∈G1⊠RkG2 be denoted by dr,s:(7)HZG1⊠RkG2=∑r1,s1r2,s2∈EG1⊠RkG2dr1,s1+dr2,s22∑rϵVG1∑s1s2ϵEG2dr,s1+dr,s22+∑sϵVG2∑r1r2ϵERkG1dr1,s+dr2,s2+∑r1r2ϵERkG1∑s1s2ϵVG2dr1,s1+dr2,s22=∑A+∑B+∑C,∑A=∑rϵVG1∑s1s2ϵEG2dr,s1+dr,s22=∑rϵVG1∑s1s2ϵEG22dr+ds1+2drds1+2dr+ds2+2drds22=∑rϵVG1∑s1s2ϵEG24dr+ds1+ds2+2drds1+ds22=∑rϵVG1∑s1s2ϵEG24d2r+1+4drd2s1+d2s2+2ds1ds2+8dr+16d2r×ds1+ds2+16d2r=HZG24M1G1+nG1+8eG1+16M1G2eG1+M1G1+16M1G1eG2OR=8eG2M1G1+nG1FG2+4M1G1FG2+8eG1M1G2+8M1G1M1G2+8eG1FG2+24M1G1eG2+M1G2+M2G2+4eG1M1G2+2M2G2+M2G2nG1,∑B=∑r1r2ϵERkG1∑sϵVG2dr1,s+dr2,s2=∑r1r2ϵERkG1r1,r2ϵVG1∑sϵVG2dr1,s+dr2,s2+∑r1r2ϵERkG1r1ϵVG1,r2ϵVRkG1−VG1∑sϵVG2dr1,s+dr2,s2+∑r1r2ϵERkG1r1,r2ϵVRkG1−VG1∑sϵVG2dr1,s+dr2,s2=∑B1+∑B2+∑B3,∑B1=∑r1r2ϵERkG1r1,r2ϵVG1∑sϵVG2dr1,s+dr2,s2=∑r1r2ϵERkG1r1,r2ϵVG1∑sϵVG2dr1+ds+dr1ds+dr2+ds+dr2ds2=∑r1r2ϵERkG1r1,r2ϵVG1∑sϵVG2dr1+dr2+2ds+dsdr1+dr22=∑r1r2ϵERkG1r1,r2ϵVG1∑sϵVG24d2r1+4d2s+4d4d2r2+8dr1ds+8dr1dr2+8dsdr2+4d2sd2r1+d2r2+2dsd2r1+8ds1d2r1+d2r2+2dsd2r1+8d2s1dr1+dr2=4eG1M1G2+4nG2+4M1G2+16eG2HZG1+16eG2M1G1+8M1G1M1G2.OR=4nG2FG1+2eG1M1G2+4M1G2FG1+8eG2M1G1+4M1G1M1G2+16eG2FG1+24M2G1nG2+4eG2+M1G2+2M1G1M1G2+2eG2+eG1M1G2,∑B2=∑r1r2ϵERkG1r1ϵVG1,r2ϵVRkG1−VG1∑sϵVG2dr1,s+dr2,s2=∑r1r2ϵERkG1r1ϵVG1,r2ϵVRkG1−VG1∑sϵVG2dr1+dr2+ds+dr1+dr2ds2=∑r1r2ϵERkG1r1ϵVG1,r2ϵVRkG1−VG1∑sϵVG22dr1+3ds+2dr1ds+22=4nG2FG1+18eG1M1G2+8eG1nG2+40eG2M1G1+48eG1eG2+8nG2M1G1+4FG1M1G2+12M1G1M1G2+16eG2FG1,∑B3=∑r1r2ϵERkG1r1,r2ϵVRkG1−VG1∑sϵVG2dr1,s+dr2,s2=∑r1r2ϵERkG1r1,r2ϵVRkG1−VG1∑sϵVG2dr1+dr1ds+dr2+dr2ds2=∑r1r2ϵERkG1r1,r2ϵVRkG1−VG1∑sϵVG24+4ds2=∑r1r2ϵERkG1r1,r2ϵVRkG1−VG1∑sϵVG216+16d2s+32ds=∑sϵVG216k−1eG11+d2s+2ds=16k−1eG1nG2+M1G2+4eG2,∑C=∑r1r2ϵERkG1∑s1s2ϵVG2dr1,s1+dr2,s22=∑r1r2ϵERkG1r1,r2ϵVG1∑s1s2ϵVG2dr1,s1+dr2,s22+∑r1r2ϵERkG1r1ϵVG1,r2ϵVRkG1−VG1∑s1s2ϵVG2dr1,s1+dr2,s22+∑r1r2ϵERkG1r1,r2ϵVRkG1−VG1∑s1s2ϵVG2dr1,s1+dr2,s22=∑C1+∑C2+∑C3,∑C1=∑r1r2ϵERkG1r1,r2ϵVG1∑s1s2ϵVG2dr1,s1+dr2,s22=∑r1r2ϵERkG1r1,r2ϵVG1∑s1s2ϵVG2dr1+ds1+dr1ds1+dr2+ds2+dr2ds22=∑r1r2ϵERkG1r1,r2ϵVG1∑s1s2ϵVG2dr1+ds1+dr1ds12+dr1+ds2+dr2ds22+24dr1dr2+2dr1ds2+dr2ds1+ds1ds2+4dr1dr2ds1ds2+4dr1dr2ds1+ds2+2dr1+dr2ds1ds2=8eG2FG1+2eG1FG2+4FG1FG2+4M1G1M1G2+4M1G1FG2+8M1G2FG1+28M2G1eG2+2M1G1M1G2+2M2G2eG1+8M2G1M2G2+M1G2+4M1G1M2G2,∑C2=∑r1r2ϵERkG1r1ϵVG1,r2ϵVRkG1−VG1∑s1s2ϵVG2dr1,s1+dr2,s22=∑r1r2ϵERkG1r1ϵVG1,r2ϵVRkG1−VG1∑s1s2ϵVG2dr1+ds1+dr1ds1+dr2+dr2ds22=∑r1r2ϵERkG1r1ϵVG1,r2ϵVRkG1−VG1∑s1s2ϵVG2dr1+ds1+dr1ds12+dr2+dr2ds22+2dr1+ds1+dr1ds1dr2+dr2ds2=8eG2FG1+32eG1eG2+4FG1FG2+4M1G1M1G2+8FG1M1G2+2eG1FG2+4M1G1FG2+8eG1FG2+16eG1M1G2+8eG1M1G2+M2G2+2M2R1G1−4M2G12eG2+2M1G2+2M2G2,∑C3=∑r1r2ϵERkG1r1,r2ϵVRkG1−VG1∑s1s2ϵVG2dr1,s1+dr2,s22=∑r1r2ϵERkG1r1,r2ϵVRkG1−VG1∑s1s2ϵVG2dr1+dr2+dr1ds1+dr2ds22=2+2ds12+2+2ds22+22+2ds12+2ds2=8k−1eG12eG2+FG2+2M1G2+16k−1eG1eG2+M1G2+M2G2.
Hence, we reached at our required result.Theorem 3.
Fork≥1, the HZ-index of G1⊠QkG2 is(8)HZG1⊠RkG2=2k−1FG1+2M2G13nG2+5M1G1+14eG2+FG2+knG2+6eG2+3M1G2+FG2M4G1−2FG1+2M2G1−4M2G1+∑uϵVG1d2u∑vϵNudv+6eG2M1G1+10eG2FG2+3FG1FG2+6M1G1M1G2+FG2nG1+3M1G1+6eG2+4M2G1+FG1nG2+7M1G2+6eG2M2G1+8M2G1eG2+M1G2+2knG2+6eG2+3M1G2+2M2G212∑uϵVG1dG14v−dG13v+∑uϵVG1tdG1udG1v+∑uϵVG1dG12v∑uϵVG1uvϵEG1dG1u−2M2G1+M2G24eG1+nG1+2eG1M1G2+M1G15e2+5M1G2+5M2G2+kM3G1+2M2G16eG2+3M1G2+2M2G2+nG2.Proof.
Let the degree of a vertexr,s∈G1⊠QkG2 be denoted by dr,s:(9)HZG1⊠QkG2=∑r1,s1r2,s2∈EG1⊠SkG2dr1,s1+dr2,s22=∑rϵVG1∑s1s2ϵEG2dr,s1+dr,s22+∑sϵVG2∑r1r2ϵEQkG1dr1,s+dr2,s2+∑r1r2ϵEQkG1∑s1r2ϵVG2dr1,s1+dr2,s22=∑A+∑B+∑C,∑A=∑rϵVG1∑s1s2ϵEG2dr,s1+dr,s22=8e1M1G2+n1HZG2+4e2M1G1+4e1HZG2+M1G1HZG2+4M1G1M1G2OR=2EH2M1H1+VH1FH2+M1H1FH2+4EH1M1H2+2M1H1M1H2+4EH1FH2+2M1G1eG2+M1G1M1G2+M2G2+2eG1M1G2+2M2G2+M2G2nG1,∑B=∑sϵVG2∑r1r2ϵEQkG1dr1,s+dr2,s2=∑sϵVG2∑r1r2ϵEQkG1r1ϵVG1,r2ϵVQkG1−VG1dr1,s+dr2,s2+∑sϵVG2∑r1r2ϵEQkG1r1,r2ϵVQkG1−VG1dr1,s+dr2,s2=∑B1+∑B2,∑B1=∑sϵVG2∑r1r2ϵEQkG1r1ϵVG1,r2ϵVQkG1−VG1dr1,s+dr2,s2=∑sϵVG2∑r1r2ϵEQkG1r1ϵVG1,r2ϵVQkG1−VG1dr1+ds+dr1dr+dr2+dr2ds2=∑r1r2ϵEQkG1s1ϵVH1s2ϵVQkH1−H1∑sϵVG2dr1+ds+dr1ds2dr2+dr2ds2+2dr1+ds+dr1dsdr2+ds+dr2ds.
Considerr1ϵVG1 and d2r1 occurs dr1 times. Thus,(10)D1=∑r1r2ϵEQG1,r1ϵVG1,r2ϵVQG1−VG1d3r1=FG1.
Let(11)D2=∑r1r2ϵEQG1,r1ϵVG1,r2ϵVQG1−VG1d2s2,as s2=uvϵEG1 and d2s2 occurs two times. Therefore,(12)D2=2∑s2=uvϵVQG1−VG1du+dv2=2∑uvϵEG1d2u+d2v+2dudv=2FG1+2M2G1,∑B1=nG2FG1+2eG1M1G2+M1G2FG1+4eG2M1G1+FG1+2M1G1M1G2+2nG2FG1+2M2G1+2M1G2FG1+2M2G1+8eG2FG1+2M2G1+2M3G1+2M2G1nG2+4eG2+M1G2+2M1G12eG2+M1G2,∑B2=∑r1r2ϵEQkG1r1r2ϵVQkG1−VG1∑sϵVG2dr1,s+dr2,s2.
Now, assumeΣB2=ΣB3+ΣB4 as follows:(13)∑B3=∑r1r2ϵEQkG1r1r2ϵVQkG1−VG1∑sϵVG2dr1+dr1ds2+dr2+dr2ds2+2dr1+dr1dsdr2+dr2ds=2k−1FG1+2M2G1nG2+M1G2+4eG2+M3G1+2M2G1nG2+4eG2+M1G2,∑B4=∑r1r2ϵEQkG1r1r2ϵVQkG1−VG1∑sϵVG2dr,s1+dr,s22=∑r1r2ϵEQkG1r1r2ϵVQkG1−VG1∑sϵVG2dr1+dr1ds+dr2+dr2ds2=∑r1r2ϵEQkG1r1,r2ϵVQkG1−G1∑sϵVG2dr12+dr22+ds2dr12+dr22+2dsdr12+dr22+2dr1+dr1dsdr2+dr2ds,D3=∑r1r2ϵEQG1r1,r2ϵVQG1−VG1d2r1+d2r2.
InD3, coefficient of(14)d2u=22dG1u+∑vϵNudv−du=d2u−2du+∑vϵNudv.
Therefore,(15)∑uϵVG1d2u=M4G1−2FG1+∑uϵVG1d2u∑vϵNudv.
For coefficient ofdudv, let r1r2ϵEQG1 with r1=uv and r2=wz. As r1r2∈EQG1, we have either v=w or z or u=w or z. So, uv is adjacent to all those vertices in G1 which are adjacent to u and v. Consequently, the number of such dudv is du+dv−2. Therefore,(16)2∑uv∈EG1dudv=2∑uvϵEG1du+dv−2dudv=2∑uv∈EG1du+dvdudv−4∑uv∈EG1dudv=2M2G1−4M2G1,so(17)D3=M4G1−2FG1+∑uϵVG1d2u∑vϵNudv+2M2G1−4M2G1,∑B4=knG2+4eG2+M1G2M4G1−2FG1+2M2G1−4M2G1+∑uϵVH1d2u∑vϵNudv+2knG2+4eG2+M1G212∑vϵVG1dG14v−dG13v+∑uvϵVG1tdG1udG1v+∑vϵVG1dG12v∑vϵVG1uvϵEG1dG1u−2M2G1,where t is the number of neighbors which are common vertices of u and v in G1.(18)∑C=∑r1r2ϵEQkG1∑s1s2ϵVG2dr1,s1+dr2,s22=∑r1r2ϵEQkG1r1ϵVG1,r2ϵVQkG1−VG1∑s1s2ϵVG2dr1,s1+dr2,s22+∑r1r2ϵEQkG1r1ϵVG1,r2ϵVQkG1−VG1∑s1s2ϵVG2dr1,s1+dr2,s22=∑C1+∑C2,∑C1=∑r1r2ϵEQkG1r1ϵVG1,r2ϵVQkG1−VG1∑s1s2ϵVG2dr1,s1+dr2,s22=∑r1r2ϵEQkG1r1ϵVG1,r2ϵVQkG1−VG1∑s1s2ϵVG2dr1+ds1+dr1ds1+dr2+ds2+dr2s22=∑r1r2ϵEQkG1r1ϵVG1,r2ϵVQkG1−VG1∑s1s2ϵVG2dr1+ds1+dr1ds12+dr2+dr2ds22+2dr1,s1dr2,s2=6eG2+M1G2FG1+3FG1FG2+2M1G1M1G2+2eG1+M1G1+2M2G1FG2+8M2G1×eG2+M1G2+2M3G1+2M2G12eG2+2M1G2+2M2G2+2M1G12M2G2+M1G2.
Now, assumeΣC2=ΣC3+ΣC4 as follows:(19)∑C3=∑s1s2ϵVG2∑r1r2ϵEQkG1r1ϵVG1,r2ϵVQkG1−VG1dr1,s1+dr2,s22=∑s1s2ϵVG2∑r1r2ϵEQkG1r1ϵVG1,r2ϵVQkG1−VG1dr1+dr1ds1+dr2+dr2ds22=∑t1t2ϵEH2∑s1s2ϵEQkH1s1,s2ϵVQkH1−H1ds1+ds1dt12+ds2+ds2dt22+2dr1+dr1dsdr2+dr2ds=2k−1FG1+2M2G12eG2+FG2+2M1G2+2k−12eG2+2M1G2+2M2G2M3G1+2M2G1,∑C4=∑s1s2ϵVG2∑r1r2ϵEQkG1r1,r2ϵVQkG1−VG1dr1,s1+dr2,s22=∑s1s2ϵVG2∑r1r2ϵEQkG1r1,r2ϵVQkG1−VG1dr1+dr2+dr1ds1+dr2ds22=∑s1s2ϵℰG2∑r1r2ϵEQkG1r1,r2ϵVQkG1−VG1dr1+dr1ds12+dr2+dr2ds22+2dr1+dr1dsdr2+dr2ds=k2eG2+FG2+2M1G2M4G1−2FG1+2M2G1−4M2G1+∑uϵVG1d2u∑vϵNudv+2k2eG2+2M1G2+2M2G212∑vϵVG1dG14v−dG13v+∑uvϵVG1tdG1udG1v+∑uϵVG1dG12v∑uϵVG1uvϵEG1dG1u−2M2G1,where t is the number of neighbors which are common vertices of u and v in G1.
Thus, we arrive at our desired result.Theorem 4.
Fork≥1, the HZ-index of G1⊠TkG2 is(20)HZG1⊠TkG2=2k−1FG1+2M2G1nG2+3M1G2+6eG2+FG2+knG2+6eG2+3M1G2+FG2M4G1−2FG1+2M2G1−4M2G1+∑uϵVG1d2u∑vϵNudv+FG1+2M2G12nG2+6M1G2+12eG2+2FG2+4FG12nG2+6M1G2+12eG2+2FG2+FG2nG1+12M1G1+12eG2+12eG1M1G2+16eG2M1G1+20M1G1M1G2+2knG2+6eG2+3M1G22M2G212∑vϵVG1dG14v−dG13v+∑uvϵVG1dG1udG1v+∑vϵVG1dG12v∑uϵVG1uvϵEG1dG1u−2M2G1+M2G14nG2+24eG2+8M2G2+12M1G2+kM3G1+2M2G16eG2+nG2+3M1G2+2M2G2+5M1G2eG1+M2G210eG1+nG1+M1G110eG2+11M1G2+10M2G2.


## 4. Applications and Discussion

Usingk=1, in Theorems 1–4, the results are obtained for the generalized D1-sum graphs as follows:(i)
S1-sum:(21)HZG1⊠S1G2=nG2+3M1G2+6eG2FG1+nG1+3M1G1+14eG1FG2+6eG2M1G1+30eG1M1G2+6M1G1M1G2+48eG1eG2+FG1FG2+8nG2eG1+2M2G1+4eG15eG2+3M1G2+2M2G2+nG2+14eG1M1G2+M2G212eG1+nG1+M1G1eG2+M1G2+M2G2+8eG1eG2.(ii)
R1-sum:(22)HZG1⊠R1G2=8nG2+6eG2FG1+nG1+20eG1FG2+8FG1FG2+24eG2M1G1+36eG1M1G2+24M1G1M1G2+24FG1M1G2+8nG2eG1+48eG1eG2+12FG2M1G1+22M1G14eG2+6M1G2+2M2G2+4eG13M1G2+3M2G2+4eG2+M2R1G1−4M2G16eG2+3M1G2+2M2G2+nG2+eG1M1G2+4M2G1nG2+6eG2+3M1G2+M2G2nG1+2eG1.(iii)
Q1-sum:(23)HZG1⊠Q1G2=2nG2+6eG2+3M1G2+2M2G212∑vϵVG1dG14v−dG13v+∑uvϵVG1tdG1udG1v+∑vϵVG1dG12v∑uϵVG1uvϵEG1dG1u−2M2G1+M2G24eG1+nG1+2eG1M1G2+M1G1×5e2+5M1G2+5M2G2+M3G1+2M2G16eG2+3M1G2+2M2G2+nG2+nG2+eG2+3M1G2+FG2M4G1−2FG1+2M2G1−4M2G1+∑uϵVG1d2u×∑vϵNudv+6eG2M1G1+10eG2FG2+3FG1FG2+6M1G1M1G2+FG2nG1+3M1G1+6eG2+4M2G1+FG1nG2+7M1G2+6eG2M2G1+8M2G1eG2+M1G2.(iv)
T1-sum:(24)HZG1⊠T1G2=nG2+6eG2+3M1G2+FG2M4G1−2FG1+2M2G1−4M2G1+∑uϵVG1d2u×∑vϵNudv+FG1+2M2G12nG2+6M1G2+12eG2+2FG2+4FG12nG2+6M1G2+12eG2+2FG2+FG2nG1+12M1G1+12eG2+12eG1M1G2+16eG2M1G1+20M1G1M1G2+2nG2+6eG2+3M1G2+2M2G212∑vϵVG1dG14v−dG13v+∑uvϵVG1tdG1udG1v+∑vϵVG1dG12v∑uϵVG1uvϵEG1dG1u−2M2G1+M3G1+2M2G1×6eG2+nG2+3M1G2+2M2G2+5M1G2eG1+M1G110eG2+11M1G2+10M2G2+M2G210eG1+nG1+M2G14nG2+24eG2+8M2G2+12M1G2.Now, we present tabular form in Table1 and graphical representation in Figure 4 of path graphs for k=1.Table 1
Hyper-Zagreb index ofF1-sum path graphs.
[n1,n2]HZPn1⊠S1Pn2HZPn1⊠R1Pn2HZPn1⊠Q1Pn2HZPn1⊠T1Pn2(3, 3)414411870883416168(4, 4)11040302602450047180(5, 5)21232621224651689230(6, 6)336969934475416145204(7, 7)51384147638110384212318Figure 4
Graphical representation ofHZPn1⊠S1Pm, HZPn1⊠R1Pm, HZPn1⊠Q1Pm, and HZPn1⊠T1Pm in red, green, orange, and purple colour, respectively.Finally, we close this section with the comment that the problem is still open for other topological indices and product of graphs, in particular the general randic index ofFk-sum graphs under corona product.



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*Source: 3304939-2021-11-20.xml*