# Eccentric Harmonic Index for the Cartesian Product of Graphs

Authors: Kamel Jebreen; Muhammad Haroon Aftab; M. I. Sowaity; B. Sharada; A. M. Naji; M. Pavithra
Journal: Journal of Mathematics
 (2022)

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## Abstract

Supposeρ is a simple graph, then its eccentric harmonic index is defined as the sum of the terms 2/ea+eb for the edges vavb, where ea is the eccentricity of the ath vertex of the graph ρ. We symbolize the eccentric harmonic index (EHI) as He=Heρ. In this article, we determine He for the Cartesian product (CP) of particularly chosen graphs. Lower bounds for He of the CP of the two graphs are established. The formulas of EHI for the Hamming and Hypercube graphs are obtained. These obtained formulas can be used in QSAR and QSPR studies to get a better understanding of their applications in mathematical chemistry.


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## Body

## 1. Introduction

Authors Throughout this study, we assume that all graphs are simple, connected as well as finite. A graphρV′,E′ is known as a simple graph if it has no loops, no directed, and multiple edges. Whereas n′=V′ρ and m′=E′ρ depict the cardinality of the graph ρ. The open neighborhood of v′ for v′∈V′ρ in the graph ρ, is symbolized by N′v′. N′v′ consists of all nodes which are adjacent to v′ and the closed neighborhood of v′ is N′v′=N′v′∪v′. The degree of a node va′ in ρ is da=dva′=N′va′. The graph ρ is said to be R-regular graph if dv′=R for all v′∈V′ρ. The shortest distance between any two connecting nodes is denoted by du′,v′ for a graph ρ. The eccentricity of a node v′∈V′ρ is ev′=maxdu′,v′:v′∈V′ρ. The radius of ρ is r′=r′ρ=minev′:v′∈V′ρ, and the diameter of ρ is diamρ=maxev′:v′∈V′ρ. Hence, r′ρ≤ev′≤diamρ for every v′∈V′ρ. In a connected graph ρ, a node v′ is a central node if ev′=r′ρ, while a node v′ in a connected graph ρ is the peripheral node if ev′=diamρ. It is important to remember that for a self-centered graph ρ, the condition ev′=r′ρ=diamρ must be true.If the graphρ is regular under the constraint ev′=r′ρ=diamρ, then ρ is familiarized as a regular self-centered graph. We depict the eccentricity of a node va′ by eva′=ea. Denoted by Pn′,Ka,b′,Cn′,Kn′,K1,n−1′ the path, complete bipartite, cycle, complete, and star graphs, respectively. The reference includes all terminologies and definitions.


## 2. Historical and Preliminary Overview

We have a set of particular fixed parameters (invariants) which have no changes for isomorphic graphs and give us some information about various factors. They are familiarized with the term “topological descriptors.” These topological descriptors (TDs) are very useful in solving many everyday life problems. They have a vital role in the graphical construction of chemical compounds. TDs have many uses in various areas such as the molecular graph theory because of their benefits over the existing experimental methods. The pharmaceutical industries paid their keen roles towards the increased day-by-day interest in TDs because of the need of human beings to low the expenditures involved in the synthesis, in vivo or in vitro clinical testing of the latest medicinal compounds. TDs can be categorized into the following three subclasses according to their properties:(i)
Formulated by means of matrices(ii)
Formulated by means of node degrees(iii)
Formulated by means of distancesThe most commonly used TDs [1, 2] are the geometric-arithmetic index, the first and second Zagreb indices, the harmonic index, the first and second multiplicative Zagreb indices, the Narumi–Katayama index, the sum-connectivity index, and the atom-bond connectivity index.During 1972–1975, Trinajstic and Gutman formulated and elaborated [3, 4] the 1st and 2nd Zagreb indices and determined by(1)M1ρ=∑i=1ndi2,M1ρ=∑vivj∈Eρdi×dj.The harmonic indexH(ρ) of a graph ρ was developed by the mathematician Fajtlowicz [5–10], and defined by(2)Hρ=∑vivj∈Eρ2di+dj.For details aboutH(ρ), the authors' advice see [11–14].In 2000, the connective eccentric indexCξ(ρ) was defined by Gupta et al. [15], as follows:(3)Cξρ=∑i=1ndiei.The eccentric harmonic indexHe(ρ) of a graph ρ was determined by Sowaity et al. [16], and formulated by(4)Heρ=∑vivj∈Eρ2ei+ej.Pavithra et al. [17] computed the EHI for the subdivision of particular graphs. Motivated by this, we compute the EHI for the CP of particular graphs.Definition 1.
For the given graphsG′ and H′, their CP [18] is symbolized by G′□H′ and determined as the graph on the vertex set VG′×VH′, and vertices u′=u1′,u2′ and v′=v1′,v2′ of VG′×VH′ are connected by an edge iff either u1′=v1′andu2′v2′∈EH′ or u2′=v2′andu1′v1′∈EG′. We know that the CP of graphs is associative as well as commutative. VG′□H′=VG′VH′, the distance between any two nodes u′=u1′,u2′ and v′=v1′,v2′ in G′□H′ is given by dG′□H′u′,v′=dG′u1′,v1′+dH′u2′,v2′. The eccentricity of the vertex u′,v′∈G′□H′ is symbolized by eu′,v′/G′□H′.Definition 2.
(1)
A Hypercube graphQr is the CP of r copies of K2(2)
The Hamming graphH′t,n is equivalent to the CP of r copies of the complete graph KnDefinition 3.
The ladder graphLn is defined as the CP of the path Pn and P2, i.e., Ln=Pn□P2.Definition 4.
The circular ladder graphCLn is formulated as the CP Cn□P2. The following results are used in proving the main and the secondary results:Observation 1.
Letm, n ∈ Z+, m, n ≥ 2. Then, m + n ≤ mn with equality holds iff m = n = 2.Lemma 1 (see [19]).
SupposeG′ and H′ are two disjoint graphs with n1,n2 vertices, respectively. Then, for any node u′,v′∈VG′□H′eu′,v′/G′□H′=eu′/G′+ev′/H′.Theorem 1.
Letρ be a graph that is self-centered [16] having order n′ and size m′, then Heρ=m/diamρ.Corollary 1 (see [16]).
LetCn be a cycle, then its EHI is HeCn=2,for evenn,2n/n−1,for oddn..


## 3. Eccentric Harmonic Index for the Cartesian Product of Graphs

We derive the EHI for the CP of some famous graphs which have a good application in Chemistry, Electronics, Electrical, and Wireless communication areas. Eccentric harmonic index for the Hamming and Hypercube Graphs are obtained.Corollary 2.
SupposeG′ and H′ are two disjoint graphs having n1,n2 vertices, respectively. Then, HeG′□H′≥n1/2diamG′HeH′+n2/2diamH′HeG′, and the above equality holds iff G′ and H′ both are complete graphs.Proof.
SupposeG′ and H′ are two disjoint graphs with n1,n2 nodes and m1,m2 edges, respectively. Then by definition of HeG′ of a graph G′ and Lemma 1, we get(5)HeG′□H′=∑wiwj∈EG′□H′2ei+ej=∑u∈VG′∑vivj∈EH′2eu,vi+eu,vj+∑v∈VH′∑uiuj∈EG′2eui,v+euj,v=∑u∈VG′∑vivj∈EH′2eu+evi+eu+evj+∑v∈VH′∑uiuj∈EG′2eui+ev+euj+ev.
From Observation1, we get(6)HeG′□H′≥∑u∈VG′∑vivj∈EH′12eu2evi+evj+∑v∈VH′∑uiuj∈EG′12ev2eui+euj,=∑u∈VG′12eu∑vivj∈EH′2evi+evj+∑v∈VH′12ev∑vivj∈EG′2eui+euj≥n12diamG′HeH′+n22diamH′HeG′,(7)=∑u∈VG′12eu∑vivj∈EH′2evi+evj+∑v∈VH′12ev∑vivj∈EG′2eui+euj≥n12diamG′HeH′+n22diamH′HeG′.
To show equality, by using inequality (7), we get that G′ must be a self-entered graph. Also, by inequality (6) we get the equality iff 2eu+evi+evj=4 and 2ev+eui+euj=4, which holds if and only if evi=evj=1 and eui=euj=1.
Thus, the result is as follows:Corollary 3.
SupposeG and H are two disjoint self-centered graphs having n1,n2 vertices, m1,m2 edges, and r1,r2 diameters, respectively. Then,(8)HeG□H≥n1m2+n2m12r1r2,with equality holds iff G and H are both complete graphs.Proof.
SupposeG and H are two disjoint graphs with n1,n2 vertices, m1,m2 edges, and r1,r2 diameters, respectively. Then by Corollary 2, we get(9)HeG□H≥n12r1HeH+n22r2HeG.
By using Theorem1, we get(10)HeG□H≥n12r1m2r2+n22r2m1r1=n1m2+n2m12r1r2.
Assume that the size ofG□H=m. Then, clearly m=n1m2+n2m1. Thus, by employing m, we can rewrite (8) as(11)HeG□H≥m2r1r2.Corollary 4.
LetKn1 and Kn2 be two disjoint complete graphs with n1,n2 vertices, respectively. Then, HeKn1□Kn2=n1n2n1+n2−2/4.Proof.
LetKn1 and Kn2 be two disjoint complete graphs with n1,n2 vertices, respectively. Then by Corollary 3, we get(12)HeKn1□Kn2=n1m2+n2m12r1r2.
SinceKn1 and Kn2 are both self-centered graphs with r1=r2=1, then(13)HeKn1□Kn2=n1n2n2−1/2+n2n1n1−1/22=n1n2n2−1+n1n2n1−14=n1n2n1+n2−24.Theorem 2.
SupposeG1,G2,…,Gt are t-disjoint graphs with orders n1,n2,…,nt, respectively. Then, the eccentricity of any vertex u1,u2,…,ut∈□i=1tGi is eu1,u2,…,ut/□i=1tGi=∑i=1teui/GiProof.
SupposeG1,G2,…,Gt are t disjoint graphs with orders n1,n2,…,nt, respectively. Assume that t = 2. Then by Lemma 1, we get(14)eu1,u2G1□G2=eu1G1+eu2G2.
Suppose it is true fort=k,2≤k≤t, i.e.,(15)eu1,u2,…,uk□i=1kGi=∑i=1keuiGi.
Then, fort=k+1, from Lemma 1 and since the Cartesian product is associative, we get(16)eu1,u2,…,uk,uk+1□i=1k+1Gi=eu1,u2,…,uk□i=1kGi+euk+1Gk+1=∑i=1keuiGi+euk+1Gk+1=∑i=1k+1euiGi.
Hence,(17)eu1,u2,…,ut□i=1tGi=∑i=1teuiGi.Theorem 3.
LetKn1,Kn2,…,Knt be t disjoint complete graphs with orders n1,n2,…,nt, respectively. Then,(18)He□i=1tKni=P2t∑i=1tni−t,where P=∏i=1tni.Proof.
LetKn1,Kn2,…,Knt be t disjoint complete graphs with orders n1,n2,…,nt, respectively. Then by the definition of HeG of a graph G, we get(19)He□i=1tKni=∑wiwj∈E□i=1tKni2ei+ej=∑u1∈VG1∑u2∈VG2…∑ut−1∈VGt−1∑uiuj∈EGt2eu1,…,ut−1,ui/□r=1tGr+eu1,…,ut−1,uj/□r=1tGr+∑u1∈VG1∑u2∈VG2…∑ut∈VGt∑uiuj∈EGt−12eu1,…,ui,ut/□r=1tGr+eu1,…,uj,ut/□r=1tGr⋮+∑u2∈VG2∑u3∈VG3…∑ut∈VGt∑uiuj∈EG12eui,u2,…,ut/□r=1tGr+euj,u2,…,ut/□r=1tGr.
By Theorem2, we get(20)He□i=1tKni=∑u1∈VG1∑u2∈VG1…∑ut−1∈VGt−1∑uiuj∈EGt1t+∑u1∈VG1∑u2∈VG1…∑ut∈VGt∑uiuj∈EGt−11t⋮+∑u2∈VG2∑u3∈VG3…∑ut∈VGt∑uiuj∈EG11t=1t∑u1∈VG1∑u2∈VG2…∑ut−1∈VGt−1ntnt−12+1t∑u1∈VG1∑u2∈VG2…∑ut∈VGtnt−1nt−1−12⋮+1t∑u2∈VG2∑u3∈VG3…∑ut∈VGtn1n1−12=1tntn1−12n1n2…nt−1+1tnt−1nt−1−12n1n2…nt−2nt+⋯+1tntn1−12n2n3…nt.
LetP=∏i=1tni. Then,(21)He□i=1tKni=1t∑i=1tPni−12=P2t∑i=1tni−t.Corollary 5.
The eccentric harmonic index of the Hamming graphHt,n is(22)HeHt,n=ntn−12.Proof.
LetHt,n be the Hamming graph. Then by substituting P=nt and ∑i=1tni=nt in (18), we get(23)HeHt,n=nt2tnt−t=nt2ttn−1=ntn−12.Corollary 6.
The eccentric harmonic index of the Hypercube graphQt is HeQt=2t−1.Proof.
LetHt,n be the Hypercube graph. Then by substituting n=2 in (22), we get(24)HeQt=2t2−12=2t−1.Theorem 4.
The eccentric harmonic index for the ladder graphLn is HeLn=1/2CξPn+2+2HePn+2−10n+9/n+12n+1.Proof.
LetLn be the ladder graph. Then by the definition of HeG, we get(25)HeLn=∑wiwj∈ELn2ei+ej=∑u∈VPn∑vivj∈EP22eu,vi+eu,vj+∑v∈VP2∑uiuj∈EPn2eui,v+euj,v.
By Lemma1,(26)∑u∈VPn∑vivj∈EP22eu,vi+eu,vj=∑u∈VPn∑vivj∈EP222eu+evi+evj=∑u∈VPn∑vivj∈EP222eu+2=∑u∈VPn1eu+1=12CξPn+2−1n−1.
Moreover,(27)∑v∈VP2∑uiuj∈EPn2eui,v+euj,v=∑v∈VP2∑uiuj∈EPn2eui+euj+2ev=∑v∈VP2∑uiuj∈EPn2eui+1+euj+1=2∑uiuj∈EPn2eui+1+euj+1=2HePn+2−22n+1+n=2HePn+2−82n+1.
By substituting (26) and (27) in (25), we get(28)HeLn=12CξPn+2−1n−1+2HePn+2−82n+1=12CξPn+2+2HePn+2−10n+9n+12n+1.Theorem 5.
The eccentric harmonic index for the circular ladder graphCLn is HeCLn=3n/n/2+1.Proof.
(29)HeCLn=∑wiwj∈ECLn2ei+ej=∑v∈VP2∑uiuj∈ECn2eui,v+euj,v+∑u∈VCn∑vivj∈EP22eu,vi+eu,vj.
By Lemma1,(30)∑v∈VP2∑uiuj∈ECn2eui,v+euj,v=2∑uiuj∈ECn22ev+eui+euj=2∑uiuj∈ECn2eui+1+euj+1.
Thus,(31)∑v∈VP2∑uiuj∈ECn2eui,v+euj,v=2HeCn+2−82diamCn+2,nis even,2HeCn+1−42diamCn+1,nis odd.
By using Corollary1, we get(32)∑v∈VP2∑uiuj∈ECn2eui,v+euj,v=2n+2diamCn+2−4diamCn+2nis even2n+1diamCn+1−2diamCn+1nis odd=2ndiamCn+2,nis even,2ndiamCn+1,nis odd.
Hence,(33)∑v∈VP2∑uiuj∈ECn2eui,v+euj,v=2ndiamCn+1.
Furthermore, we have(34)∑u∈VCn∑vivj∈EP22eu,vi+eu,vj=∑u∈VCn∑vivj∈EP222eu+evi+evj=∑u∈VCn1eu+1=∑u∈VCn1diamCn+1=ndiamCn+1.
By substituting (33) and (34) in (29), we get(35)HeCLn=2ndiamCn+1+ndiamCn+1=3ndiamCn+1=3nn/2+1.


## 4. Importance of the Work

The EHI has recently been projected in a comparable way by swapping the vertex degrees with the vertex eccentricities. One of the main topics in the chemical graph theory is to study how certain invariants of product graphs are related to the corresponding invariants of their components. Due to this, here we investigate the behavior of the eccentric version of the harmonic index under various families of graph products and apply the derived results to some graphs of chemical and general interest.


## 5. Hypothesis and Limitations of the Industrialized Formulas

There are some partial limitations of this work which are as follows:(1)
These formulas are incompletely characterized(2)
There is limited knowledge of chemistry for the molecular structures of Hamming and Hypercube graphs(3)
We are constrained to hypothesisnth formula to calculate topological indices which can be utilized to interlink with the molecular compound


## 6. Conclusion

We have discussed and developed some important formulas. These obtained formulas of EHI for the Hamming and Hypercube graphs can be correlated with the physical as well as biological properties to get a better understanding of the topology of given molecular compounds.


### 6.1. Future Work

It is an open problem to apply the eccentric harmonic index in other fields such as Physics and Computer science.


## 6.1. Future Work

It is an open problem to apply the eccentric harmonic index in other fields such as Physics and Computer science.



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