# Sharp Bounds for the Inverse Sum Indeg Index of Graph Operations

Authors: Anam Rani; Muhammad Imran; Usman Ali
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Vukičević and Gasperov introduced the concept of 148 discrete Adriatic indices in 2010. These indices showed good predictive properties against the testing sets of the International Academy of Mathematical Chemistry. Among these indices, twenty indices were taken as beneficial predictors of physicochemical properties. The inverse sum indeg index denoted byISIGk of Gk is a notable predictor of total surface area for octane isomers and is presented as ISIGk=∑gkgk′∈EGkdGkgkdGkgk′/dGkgk+dGkgk′, where dGkgk represents the degree of gk∈VGk. In this paper, we determine sharp bounds for ISI index of graph operations, including the Cartesian product, tensor product, strong product, composition, disjunction, symmetric difference, corona product, Indu–Bala product, union of graphs, double graph, and strong double graph.


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## Body

## 1. Introduction

LetGk be a connected and simple graph whose vertex and edge sets are VGk and EGk, respectively. The order k and size k′ of Gk are the cardinalities of VGk and EGk, respectively. The degree formula of gk∈VGk is the cardinality of linked vertices to gk in Gk and represented by dGkgk. The largest (or smallest) degree of Gk is the degree of a vertex of Gk with the greatest (or least) number of edges incident to it and represented by ΔGk (or δGk).A molecular descriptor is a numerical parameter of a graph that distinguished its topology. In organic chemistry, topological descriptors have investigated many applications in pharmaceutical drug design, QSAR/QSPR study, chemical documentation, and isomer discrimination. Some of these topological indices are Wiener index, Zagreb indices, Szeged index, and Randić index. The set of 148 discrete Adriatic descriptors [1] have been defined in 2010. These descriptors showed well predictive characteristics on the testing sets given by International Academy of Mathematical Chemistry. Twenty of these descriptors were taken as noteworthy predictors of physicochemical properties. One such index is inverse sum indeg index, denoted by ISIGk, of Gk that was investigated in [1] as a noteworthy predictor of total surface area for octane isomers and is presented as follows:(1)ISIGk=∑gkgk′∈EGkdGkgkdGkgk′dGkgk+dGkgk′.Sedlar et al. [2] investigated graph-theoretical characteristics of ISI index. Falahati-Nezhad et al. [3] computed some sharp bounds of inverse sum indeg (ISI) index.The Zagreb indices ofGk are presented by Gutman and Trinajstić [4] as follows:(2)M1Gk=∑gk∈VGkdGkgk2,M2Gk=∑gkgk′∈EGkdGkgkdGkgk′.LetGk be k-vertex and Hl be l-vertex graphs with size k′ and l′, respectively. The Cartesian product Gk⊳Hl, whose vertex set is VGk×VHl and gk,hl and gk′,hl′ are adjacent when gk=gk′ and hlhl′∈EHl or gkgk′∈EGk and hl=hl′, is a graph. The order and size of Gk⊳Hl are kl and k′l+kl′, respectively. The degree formula for gk,hl∈VGk⊳Hl is dGkgk+dHlhl.The tensor productGk×Hl, whose set of vertices is VGk×VHl and gk,hl and gk′,hl′ are linked when gkgk′∈EGk and hlhl′∈EHl, is a graph. The order and size of Gk×Hl are kl and 2k′l′, respectively. The degree formula for gk,hl in Gk×Hl is dGkgkdHlhl.The strong productGk⊠Hl, whose vertex set and edge set are VGk×VHl and EGk⊳Hl∪EGk×Hl, respectively, is a graph. The order and size of Gk⊠Hl are kl and kl′+lk′+2k′l′, respectively. The degree formula for gk,hl in Gk⊠Hl is dGkgk+dHlhl+dGkgkdHlhl.The compositionGkHl, whose vertex set VGk×VHl and gk,hl and gk′,hl′ are linked when gkgk′∈EGk or gk=gk′ and hlhl′∈EHl, is a graph. The order and size of GkHl are kl and k′l2+kl′, respectively. The degree formula for gk,hl in GkHl is ldGkgk+dHlhl.The disjunctionGk∨Hl, whose vertex set is VGk×VHl and gk,hl and gk′,hl′ are linked when gkgk′∈EGk or hlh∈′EHl is a graph. The order and size of Gk∨Hl are kl and k′l2+l′k2−2k′l′, respectively. The degree formula for gk,hl in Gk∨Hl is ldGkgk+kdHlhl−dGkgkdHlhl.The symmetric differenceGk⊕Hl is a graph with vertex set VGk×VHl and gk,hlgk′,hl′∈EGk⊕Hl whenever gkgk′∈EGk or hlhl′∈EHl but not both. The order and size of Gk⊕Hl are kl and k′l2+l′k2−4kl, respectively. The degree formula for gk,hl∈VGk⊕Hl is ldGkgk+kdHlhl−2dGkgkdHlhl.LetGk1,Gk2,…,Gkn be all vertex disjoint graphs. Then, their join is a graph whose vertex set is ∪s=1nVGks and edge set is ∪s=1nEGks together with the edges linking VGk1 and VGk2, VGk2 and VGk3 so on VGkn−1 and VGkn. The degree formula of gk∈VGk1+Gk2+⋯+Gkn is dGksgk+r−ks, s=1,2,…,n and r=k1+k2+⋯+kn.The corona productGk∘Hl is acquired by taking Gk as a single copy and k copies of Hl and by linking r-th vertex of Gk to every vertex of r-th copy of Hl, where 1≤r≤k. The graph Gk∘Hl has size and order k′+kl′+kl and k1+l, respectively. The degree formula of g∈VGk∘Hl is(3)dGk∘Hlg=dGkg+l,forg∈VGk,dHlg+1,forg∈VHl.The Indu–Bala productGk▼Hl is obtained from two disjoint copies of Gk+Hl by linking the corresponding vertices of two copies of Hl. The order and size of Gk▼Hl are 2k+l and 2k′+2l′+2kl+l, respectively. The degree of g∈VGk▼Hl is(4)dGk▼Hlg=dGkg+l,forg∈VGk,dHlg+k+1,forg∈VHl.The double graphDGk is acquired by taking original edge set of two copies V1Gk and V2Gk of VGk and linking each vertex in V1Gk with the linked vertices of corresponding vertex in V2Gk. The strong double graph SDGk is acquired by taking two copies of V1Gk and V2Gk of VGk and linking each vertex in V1Gk with closed neighborhood of corresponding vertex in V2Gk.Figure1 depicts some graph operations. For more details on these graph operations, see [5–14]. Also, we refer some recent articles [15–19] on different kinds of descriptors.It is an important and well-reputed problem to study and explore the molecular topological descriptors of the graph operations in terms of the original graphs, say Gk and Hl, and this also helps to explore the physicochemical properties of the complex chemical structures which arise from these graph operations. The upper and lower bounds of any molecular descriptors are the important information related to a chemical graph. They determine the approximate possible range of the invariant in the form of molecular structural parameters. There are some bounds already available for the inverse sum indeg (ISI) index regarding the number of pendant vertices, size, radius, smallest and largest vertex degrees, and smallest nonpendent vertex degree of a graph computed in [3]. The objective of this article is to determine the bounds for inverse sum indeg index of some graph operations including Cartesian product, tensor product, strong product, composition, disjunction, symmetric difference, corona product, Indu–Bala product, union of graphs, double graph, and strong double graph in the form of original graphs, say Gk and Hl.Figure 1
Graph operations: (a)P3□P4; (b) P3×P4; (c) P3⊠P4; (d) P4P3; (e) P4∘P3; (f) P3∨P4; (g) P3⊕P4; (h) P3▼P4; (i) DP4; (j) SDP4.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)


## 2. Applications of Graph Theory Concept and Topological Indices in Chemistry

In 1936, Hosoya introduced the concept of graph terminologies in chemistry and provided a modeling for molecules. This modeling contents lead to predict the chemical properties of molecules, easy classification of chemical compounds, computer simulations, and computer-assisted design of new chemical compounds. As in current century, chemists manipulate graphs on a daily basis using Table1 terminologies for recent development in their research.Table 1
Graph theory and chemistry dictionary.
Graph theoryChemistryGraphStructural formulaVertexAtomEdgeChemical bondVertex degreeValency of atomTreeAcyclic structureBipartite graphAlternant structurePerfect matchingKekule structureAdjacency matrixHuckel matrixGraph hypothesis had investigated an interesting exercise around in research. Compound graph speculation has provided a collection of beneficial indices, for instance, topological indices. The Zagreb indices are the topological indices that are correlated to a substantial computation of fabricated characteristics of the particles and have been investigated parallel to establishing the Kovats constants and limit of the particles [20]. The hyper Zagreb descriptor has a strong bound between the security of direct dendrimers besides the expanded medication stores and for establishing the strain criticalness of cyclo alkanes [21]. To connect with various physico-mix characteristics, Zagreb indices have required deep control upon the essentialness of the dendrimers [22]. The Zagreb polynomials were determined to happen for computation of the π-electron imperativeness of the particles inside specific brutal verbalizations [23, 24].


## 3. Inverse Sum Indeg Index of Graph Operations

In this section, we compute the inverse sum indeg index of the Cartesian product, tensor product, strong product, composition, disjunction, symmetric difference, corona product, Indu–Bala product, double graph, and strong double graph. The relation between largest and smallest degree ofGk to the degree of gk∈VGk is as follows:(5)dGkgk≤ΔGk,dGkgk≥δGk.In the upcoming theorem, we calculate the bounds for inverse sum indeg (ISI) index of Cartesian product.Theorem 1.
LetGk and Hl be two graphs. Then,(6)M2Gk⊳Hl2ΔGk+ΔHl≤ISIGk⊳Hl≤M2Gk⊳Hl2δGk+δHl.
The equalities hold if and only ifGk and Hl are regular.Proof.
Using the degree formula for a vertex ofGk⊳Hl in equation (1),(7)ISIGk⊳Hl=∑gk,hlgk′,hl′∈EGk⊳HldGk⊳Hlgk,hldGk⊳Hlgk′,hl′dGk⊳Hlgk,hl+dGk⊳Hlgk′,hl′=∑gk,hlgk′,hl′∈EGk⊳HldGk⊳Hlgk,hldGk⊳Hlgk′,hl′dGkgk+dHlhl+dGkgk′+dHlhl′≤12δGk+δHl∑gk,hlgk′,hl′∈EGk⊳HldGk⊳Hlgk,hldGk⊳Hlgk′,hl′=M2Gk⊳Hl2δGk+δHl.
Similarly, we can evaluate(8)ISIGk⊳Hl≥M2Gk⊳Hl2ΔGk+ΔHl.
The above equalities hold if and only if factor graphs are regular.
In the next theorem, we calculate the bounds for ISI index of tensor product ofGk and Hl.Theorem 2.
LetGk and Hl be two graphs. Then,(9)M2GkM2HlΔGkΔHl≤ISIGk×Hl≤M2GkM2HlδGkδHl.
The above equalities hold if and only if both graphs are regular.Proof.
Using the degree formula for a vertex in tensor product of graphs in (1),(10)ISIGk×Hl=∑gk,hlgk′,hl′∈EGk×HldGk×Hlgk,hldGk×Hlgk′,hl′dGk×Hlgk,hl+dGk×Hlgk′,hl′=∑gk,hlgk′,hl′∈EGk×HldGk×Hlgk,hldGk×Hlgk′,hl′dGkgkdHlhl+dGkgk′dHlhl′≤12δGkδHl∑gk,hlgk′,hl′∈EGk×HldGk×Hlgk,hldGk×Hlgk′,hl′=M2Gk×Hl2δGkδHl=M2GkM2HlδGkδHl.
See Theorem 2.1 in [25]. Similarly, we can compute(11)ISIGk×Hl≥M2GkM2HlΔGkΔHl.
The above equalities hold if and only if factor graphs are regular.
We derive the bounds of inverse sum indeg (ISI) index ofGk⊠Hl in the upcoming theorem.Theorem 3.
LetGk and Hl be two graphs. Then,(12)M2Gk⊠Hl2ΔGk+ΔHl+ΔGkΔHl≤ISIGk⊠Hl≤M2Gk⊠Hl2δGk+δHl+δGkδHl.
The equalities hold if and only if both graphs are regular.Proof.
Using the degree formula of a vertex in strong product of graphs in (1),(13)ISIGk×Hl=∑gk,hlgk′,hl′∈EGk⊠HldGk⊠Hlgk,hldGk⊠Hlgk′,hl′dGk⊠Hlgk,hl+dGk⊠Hlgk′,hl′=∑gk,hlgk′,hl′∈EGk⊳HldGk⊠Hlgk,hldGk⊠Hlgk′,hl′dGkgk+dHlhl+dGkgkdHlhl+dGkgk′dHlhl′+dGkgk+dHlhl′≤12δGk+δHl+δGkδHl∑gk,hlgk′,hl′∈EGk⊳HldGk⊠Hlgk,hldGk⊠Hlgk′,hl′=M2Gk×Hl2δGk+δHl+δGkδHl.
In a similarly way,(14)ISIGk⊠Hl≥M2Gk⊠Hl2ΔGk+ΔHl+ΔGkΔHl.
The above equalities satisfy if and only if factor graphs are regular.
In the upcoming theorem, we evaluate the bounds for inverse sum indeg (ISI) index ofGkHl.Theorem 4.
LetGk and Hl be two graphs. Then,(15)M2GkHl2lΔGk+ΔHl≤ISIGkHl≤M2GkHl2lδGk+δHl.
The equalities hold if and only if both graphs are regular.Proof.
Using the degree formula of an element ofVGkHl in (1),(16)ISIGkHl=∑gk,hlgk′,hl′∈EGkHldGkHlgk,hldGkHlgk′,hl′dGkHlgk,hl+dGkHlgk′,hl′=∑gk,hlgk′,hl′∈EGkHldGkHlgk,hldGkHlgk′,hl′ldGkgk+dHlhl+ldGkgk′+dHlhl′≤12lδGk+δHl∑gk,hlgk′,hl′∈EGkHldGkHlgk,hldGkHlgk′,hl′=M2GkHl2lδGk+δHl.
In a similar way,(17)ISIGkHl≥M2GkHl2lΔGk+ΔHl.
The above equalities hold if and only if factor graphs are regular.
In the following theorem, we present the bounds for inverse sum indeg (ISI) index of disjunction ofGk and Hl.Theorem 5.
LetGk and Hl be two graphs. Then,(18)M2Gk∨Hl2lΔGk+kΔHl−ΔGkΔHl≤ISIGk∨Hl≤M2Gk∨Hl2lδGk+kδHl−δGkδHl. The equalities hold when factor graphs are regular.Proof.
Using the degree formula of an element ofVGk∨Hl in (1),(19)ISIGk∨Hl=∑gk,hlgk′,hl′∈EGk∨HldGk∨Hlgk,hldGk∨Hlgk′,hl′dGk∨Hlgk,hl+dGk∨Hlgk′,hl′=∑gk,hlgk′,hl′∈EGk∨HldGk∨Hlgk,hldGk∨Hlgk′,hl′ldGkgk+kdHlhl−dGkgkdHlhl+ldGkgk′+kdHlhl′−dGkgk′dHlhl′≤12lδGk+kδHl−δGkδHl∑gk,hlgk′,hl′∈EGk∨HldGk∨Hlgk,hldGk∨Hlgk′,hl′=M2Gk∨Hl2lδGk+kδHl−δGkδHl.
Similarly, we compute(20)ISIGk∨Hl≥M2Gk∨Hl2lΔGk+kΔHl−ΔGkΔHl.
The above equalities hold when both graphs are regular.
Next, we derive the bounds of inverse sum indeg (ISI) index ofGk⊕Hl.Theorem 6.
LetGk and Hl be two graphs. Then,(21)M2Gk⊕Hl2lΔGk+kΔHl−2ΔGkΔHl≤ISIGk⊕Hl≤M2Gk⊕Hl2lδGk+kδHl−2δGkδHl.
The equalities hold if and only if factor graphs are regular.Proof.
Using the degree formula of a vertex ofVGk⊕Hl in (1),(22)ISIGk⊕Hl=∑gk,hlgk′,hl′∈EGk⊕HldGk⊕Hlgk,hldGk⊕Hlgk′,hl′dGk⊕Hlgk,hl+dGk⊕Hlgk′,hl′=∑gk,hlgk′,hl′∈EGk⊕HldGk⊕Hlgk,hldGk⊕Hlgk′,hl′ldGkgk+kdHlhl−2dGkgkdHlhl+ldGkgk′+kdHlhl′−2dGkgk′dHlhl′≤12lδGk+kδHl−2δGkδHl∑gk,hlgk′,hl′∈EGk⊕HldGk⊕Hlgk,hldGk⊕Hlgk′,hl′=M2Gk⊕Hl2lδGk+kδHl−2δGkδHl.
Similarly,(23)ISIGk⊕Hl≥M2Gk⊕Hl2lΔGk+kΔHl−2ΔGkΔHl.The above equalities hold if and only if both graphs are regular.
Next, we evaluate the bounds of inverse sum indeg (ISI) index of join ofn graphs.Theorem 7.
LetGk=Gk1+Gk2+⋯+Gkn. Then,(24)∑s=1nM2Gks+r−ksM1Gks+ks′r−ks22ΔGks+r−ks+12∑s≠j,s,j=1n2ks′+ksr−ks2kj′+kjr−kjΔGks+ΔGkj+2r−ks−kj≤ISIGk≤∑s=1nM2Gks+r−ksM1Gks+ks′r−ks22δGks+r−ks+12∑i≠j,s,j=1n2ks′+ksr−ks2kj′+kjr−kjδGks+δGkj+2r−ks−kj.
The equalities hold if and only ifGks, for s=1,2,…,n, are regular graphs.Proof.
We assume thatVGks=ks, EGks|=ks′ for s=1,2,…,n and r=k1+k2+⋯+kn. By using the degree formula of a vertex in Gk given in (1),(25)ISIGk=∑gkgk′∈EGkdGkgkdGkgk′dGkgk+dGkgk′=∑s=1n∑gkgk′∈EGksdGksgk+r−ksdGksgk′+r−ksdGksgk+dGksgk′+2r−2ks+12∑s≠j,s,j=1n∑gk∈VGks∑gk′∈VGkjdGksgk+r−ks−dGkjgk′+r−kjdGksgk+dGkjgk′+2r−ks−kj≤∑s=1n∑gkgk′∈EGksdGksgk+r−ksdGksgk′+r−ksδGks+δGks+2r−2ks+12∑s≠j,s,j=1n∑gk∈VGks∑v∈VGkjdGksgk+r−ksdGkjgk′+r−kjδGks+δGkj+2r−ks−kj=∑s=1nM2Gks+r−ksM1Gks+ks′r−ks22δGks+r−ks+12∑s≠j,s,j=1n2ks′+ksr−ks2kj′+kjr−kjδGks+δGkj+2r−ks−kj.
Similarly(26)ISIGk≥∑s=1nM2Gks+r−ksM1Gks+ks′r−ks22ΔGks+r−ks+12∑s≠j,s,j=1n2ks′+ksr−ks2kj′+kjr−kjΔGks+ΔGkj+2r−ks−kj.
The above equalities hold if and only ifGks, s=1,2,…,n, are regular.
In the following theorem, we calculate the bounds for ISI index ofGk∘Hl.Theorem 8.
LetGk and Hl be k-vertex and l-vertex graphs. Then,(27)kM2Hl+M1Hl+l2ΔHl+1+2l′+l2k′+klΔGk+ΔHl+l+1+M2Gk+lM1Gk+l2l′2ΔGk+l≤ISIGk∘Hl≤kM2Hl+M1Hl+l2δHl+1+2l′+l2k′+klδGk+δHl+l+1+M2Gk+lM1Gk+l2l′2δGk+l.
The equalities hold if and only if both graphs are regular.Proof.
Using the degree formula of a vertex in corona product in (1),(28)ISIGk∘Hl=k∑hlhl′∈EHldHlhl+1dHlhl′+1dHlhl+dHlhl′+2+∑s=1k∑j=1ldHlhlj+1dGkgks+ldHlhlj+dGkgks+l+1+∑gkgk′∈EGkdGkgk+ldGkgk′+ldGkgk+dGkgk′+2l.
From equation (2), we obtain(29)ISIGk∘Hl≤k∑hlhl′∈EHldHlhl+1dHlhl′+12δHl+1+∑s=1k∑j=1ldHlhlj+1dGkgks+lδGk+δHl+l+1+∑gkgk′∈EGkdGkgk+ldGkgk′+l2δGk+l=kM2Hl+M1Hl+l2δHl+1+2l′+l2k′+klδGk+δHl+l+1+M2Gk+lk′Gk+l2l′2δGk+l.
Similarly, we calculate(30)ISIGk∘Hl≥kM2Hl+M1Hl+l2ΔHl+1+2l′+l2k′+klΔGk+ΔHl+l+1+M2Gk+lM1Gk+l2l′2ΔGk+l.
The above equalities hold only whenGk and Hl are regular graphs.
Next, we evaluate the bounds for inverse sum indeg (ISI) index of Indu–Bala product.Theorem 9.
LetGk and Hl be k-vertex and l-vertex graphs. Then,(31)M2Gk+lM1Gk+l2k′ΔGk+l+2M2Hl+2l+3M1Hl+2l′+lk+12+4l′k+12ΔHl+k+1+24k′l′+2k′lk+1+2l′kl+l2kk+1ΔGk+ΔHl+k+l+1≤ISIGk▼Hl≤M2Gk+lM1Gk+l2k′δGk+l+2M2Hl+2l+3M1Hl+2l′+lk+12+4l′k+12δHl+k+1+24k′l′+2k′lk+1+2l′kl+l2kk+1δGk+δHl+k+l+1.
The equalities hold only whenGk and Hl are regular.Proof.
Using the degree formula of a vertex in Indu–Bala product in (1),(32)ISIGk▼Hl=2∑gkgk′∈EGkdGkgk+ldGkgk′+ldGkgk+dGkgk′+2l+∑hlhl′∈EHldHlhl+k+1dHlhl′+k+1dHlhl+dHlhl′+2k+2+∑gk∈VGk∑hl∈VHldGkgk+ldHlhl+k+1dGkgk+dHlhl+k+l+1+∑hl∈VHldHlhl+k+122dHlhl+k+1.
Using equation (2), then we have(33)ISIGk▼Hl≤2∑gkgk′∈EGkdGkgk+ldGkgk′+l2δGk+l+∑hlhl′∈EHldHlhl+k+1dHlhl′+k+12δHl+k+1+∑gk∈VGk∑hl∈VHldGkgk+ldHlhl+k+1δGk+δHl+k+l+1+∑hl∈VHldHlhl+k+122δHl+k+1=M2Gk+lM1Gk+l2k′δGk+l+2M2Hl+2l+3M1Hl+2l′+lk+12+4l′k+12δHl+k+1+24k′l′+2k′lk+1+2l′kl+l2kk+1δGk+δHl+k+l+1.
Similarly, we calculate(34)ISIGk▼Hl≥M2Gk+lM1Gk+l2k′ΔGk+l+2M2Hl+2l+3M1Hl+2l′+lk+12+4l′k+12ΔHl+k+1+24k′l′+2k′lk+1+2l′kl+l2kk+1ΔGk+ΔHl+k+l+1.
The equalities hold only whenGk and Hl are regular graphs.
In the next theorem, we find the inverse sum indeg (ISI) index of double graph.Theorem 10.
LetGk be a k-vertex graph. Then,(35)ISIDGk=8ISIGk.Proof.
Using the degree formula of a vertex inDGk in equation (1), we acquire(36)ISIDGk=∑gkgk′∈EDGkdDGkgkdDGkgk′dDGkgk+dDGkgk′=4∑gkgk′∈EGk2dGkgk2dGkgk′2dGkgk+2dGkgk′=8∑gkgk′∈EGkdGkgkdGkgk′dGkgk+dGkgk′=8ISIGk.
In the upcoming theorem, we calculate the bounds for inverse sum indeg (ISI) index of strong double graph.Theorem 11.
LetGk be an k-vertex graph. Then,(37)M2SDGk22ΔGk+1≤ISISDGk≤M2SDGk22δGk+1.
The equalities hold only whenGk is regular.Proof.
Using the degree formula of a vertex inSDGk in (1),(38)ISISDGk=∑gkgk′∈ESDGkdSDGkgkdSDGkgk′dSDGkgk+dSDGkgk′=∑gkgk′∈ESDGkdSDGkgkdSDGkgk′2dGkgk+1+2dGkgk′+1≤∑uv∈ESDGkdSDGkgkdSDGkgk′2δGk+1=M2SDGk22δGk+1.
Similarly, we compute(39)ISISDGk≥M2SDGk22ΔGk+1.
The above equalities hold only whenGk is a regular graph.


## 4. Conclusion

In this paper, some graph operations including different products, differences, union of graphs, double graph, and strong double graph are studied. In particular, we have found the sharp bounds for inverse sum indeg (ISI) index of these operations of graphs. The investigation related to other significant predictors is still open.



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