# Degree-Based Topological Indices of Generalized Subdivision Double-Corona Product

Authors: Ying Wang; Muhammad Waheed; Muhammad Kamran Jamil; Syed Mazhar Shah; Adnan Aslam; Hamideh Aram
Journal: Journal of Chemistry
 (2022)

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## Abstract

Graph operations play an important role in constructing complex network structures from simple graphs. Computation of topological indices of these complex structures via graph products is an important task. In this paper, we generalized the concept of subdivision double-corona product of graphs and investigated the exact expressions of the first and second Zagreb indices, first reformulated Zagreb index, and forgotten topological index (F-index) of this graph operation.


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## Body

## 1. Introduction

In mathematics [1, 2], graph theory is the study of graphs which are mathematical structures used to model pairwise connection between objects. Graph theory presents efficient and powerful tools for the topological characterization of the underline structure. Any problem which includes graph structure can be solved by using the graph theoretical approach. This enables researchers to apply graph theory in various fields such as software engineering, biology, chemistry, and operation research.We start by defining some basic notions related to graph theory. Letϒ be a simple connected graph with vertex set Vϒ and edge set Eϒ. The number of vertices and number of edges of ϒ are called its order and size, respectively. A graph of order r and size s will be denoted by ϒr,s. For any vertex p∈Vϒ, the degree of vertex p is the number of edges incident on the vertex p, and it is written as dϒp or simply dp. The subdivision of the graph ϒ is denoted by Sϒ and is obtained by inserting a new vertex on every edge of ϒ. For more details on the basic terminologies related to graph theory, we suggest readers to see the book by Gross et al. [3].Topological indices are numerical numbers assigned to a molecular graph. These numbers are invariant under graph isomorphism. Topological index is also called a molecular structure descriptor or graph theoretical descriptor [4]. One of the important uses of topological indices is in the development of quantitative structure-activity relationships (QSARs) in which the biological activity or other properties of molecules are correlated with their chemical structure. Topological indices are divided into three categories: degree based, distance based, and counting related. Among these, the most studied ones are degree-based topological indices. The first and second Zagreb indices of a graph ϒ are defined as(1)M1ϒ=∑p∈Vϒdϒp2=∑pq∈Eϒdϒp+dϒq,M2ϒ=∑pq∈Eϒdϒpdϒq.In 1972, these topological indices were applied for the first time to find the totalπ electron energy of molecular graphs [5]. Later, the Zagreb indices developed important applications in QSPR/QSAR studies, and a lot of research papers have been published on these [6–11].Miličević et al. [12] reformulated the Zagreb indices in terms of edge degree which are defined as(2)EM1ϒ=∑e∈Eϒde2,EM2ϒ=∑e∼fdedf,where de denotes the degree of an edge e=pq∈ϒ and is defined as de=dp+dq−2. The notation e∼f shows that the edges e and f are adjacent to each other.The forgotten topological index orF-index is defined as [13, 14](3)Fϒ=∑p∈Vϒdϒp3=∑pq∈Eϒdϒp2+dϒq2.In 2013, Shirdel et al. [15] proposed a new version of Zagreb indices called the first hyper Zagreb index and is defined as(4)HM1ϒ=∑pq∈Eϒdϒp+dϒq2.A large number of the topological indices have been discovered and intend to demonstrate substance, drug, and different properties of atoms.Letϒr,s,ϒ1r1,s1, and ϒ2r2,s2 be three graphs. The subdivision double-corona product of ϒ,ϒ1, and ϒ2 is denoted by ϒS∘ϒ1,ϒ2 and is obtained by taking one copy of Sϒ, r copies of ϒ1, and s copies of ϒ2 and joining the i-th old vertex of Sϒ to every vertex of r1 of the i-th copy of ϒ1 and j-th new vertex of Sϒ to every vertex of r2 of the j-th copy of ϒ2 [16]. In the following, we generalize the subdivision double-corona product. Let ϒr,s and Sϒr′,s′ be two simple connected graphs. It is supposed that we have two further simple connected graphs ϒ1r1,s1 and ϒ2r2,s2. Let X⊆Vϒ1 and Y⊆Vϒ2; then, the generalized subdivision double corona is denoted by ϒS∘ϒ1X,ϒ2Y and is obtained by taking one copy of Sϒ, r copies of ϒ1, and s copies of ϒ2 and then by joining the i-th old vertex of Sϒ to every vertex of X of the i-th copy of ϒ1 and j-th new vertex of Sϒ to every vertex of Y of the j-th copy of ϒ2. Figure 1 illustrates the definition by considering X=2,3 and Y=a.Figure 1
Generalized subdivision double coronaϒS∘ϒ1X,ϒ2Y.The graph operation, especially graph products, plays a significant role not only in pure and applied mathematics but also in computer science. For instance, the Cartesian product provides a significant model for connecting computers. In order to synchronize the work of the entire framework, it is necessary to search for Hamiltonian paths and cycles in the network [17]. Some graph operations and their topological indices are presented in [18–24].


## 2. Main Results

In this section, we present the main results. Llemma1 is useful to obtain the main results and can be directly obtained from the definition of the generalized subdivision corona product.Lemma 1.
Letϒr,s and Sϒr′,s′ be two simple connected graphs. Furthermore, it is supposed that we have two simple connected graphs, ϒ1r1,s1 and ϒ2r2,s2. Let X⊆Vϒ1 and Y⊆Vϒ2; then, the degree of vertices in the graph ϒS∘ϒ1X,ϒ2Y is(5)dϒS∘ϒ1X,ϒ2Yp=dϒp+r1′,ifp∈Vϒ,dϒ1p+1,ifp∈X,dϒ1p,ifp∉X,dϒ2p+1,ifp∈Y,dϒ2p,ifp∉Y,2+r2′,ifp∈VSϒ,where r1′=X and r2′=Y.Theorem 2.
Letϒr,s be a graph and Sϒr′,s′ be its subdivision graph. Furthermore, it is supposed that ϒ1r1,s1 and ϒ2r2,s2 are two simple graphs. We assume X⊆Vϒ1 and Y⊆Vϒ2; then, the first Zagreb index of ϒS∘ϒ1X,ϒ2Y is given as(6)M1ϒS∘ϒ1X,ϒ2Y=M1ϒ+rr12′+4sr1′+rM1ϒ1+rr1′+2r∑p∈Xdϒ1p+sM1ϒ2+sr2′+2s∑p∈Ydϒ2p+s2+r2′2.Proof.
From the definition of the first Zagreb index, we have(7)M1ϒS∘ϒ1X,ϒ2Y=∑p∈VϒS∘ϒ1X,ϒ2YdϒS∘ϒ1X,ϒ2Yp2.
Applying Lemma1, we get(8)M1ϒS∘ϒ1X,ϒ2Y=∑p∈Vϒdϒp+r1′2+r∑p∈Xdϒ1p+12+r∑p∉Xdϒ1p2+s∑p∈Ydϒ2p+12+s∑p∉Ydϒ2p2+∑p∈VSϒ2+r2′2,=∑p∈Vϒdϒp2+r1′2+2r1′dϒp+r∑p∈Xdϒ1p2+1+2dϒ1p+r∑p∉Xdϒ1p2+s∑p∈Ydϒ2p2+1+2dϒ2p+s∑p∉Ydϒ2p2+∑p∈VSϒ2+r2′2,=∑p∈Vϒdϒp2+r1′2∑p∈Vϒ1+2r1′∑p∈Vϒdϒp+r∑p∈Xdϒ1p2+r∑p∈X1+2r∑p∈Xdϒ1p+r∑p∉Xdϒ1p2+s∑p∈Ydϒ2p2+s∑p∈Y1+2s∑p∈Ydϒ2p+s∑p∉Ydϒ2p2+2+r2′2∑p∈VSϒ1,=∑p∈Vϒdϒp2+r1′2∑p∈Vϒ1+2r1′∑p∈Vϒdϒp+r∑p∈Xdϒ1p2+∑p∉Xdϒ1p2+r∑p∈X1+2r∑p∈Xdϒ1p+s∑p∈Ydϒ2p2+∑p∉Ydϒ2p2+s∑p1+2s∑p∈Ydϒ2p+2+r2′2∑p∈VSϒ1,=∑p∈Vϒdϒp2+r1′2∑p∈Vϒ1+2r1′∑p∈Vϒdϒp+r∑p∈Vϒ1dϒ1p2+r∑p∈X1+2r∑p∈Xdϒ1p+s∑p∈Vϒ2dϒ2p2+s∑p∈Y1+2s∑p∈Ydϒ2p+2+r2′2∑p∈VSϒ1,=M1ϒ+rr1′2+4sr1′+rM1ϒ1+rr1′+2r∑p∈Xdϒ1p+sM1ϒ2+sr2′+2s∑p∈Ydϒ2p+s2+r2′2,which is our required result.The classical subdivision double-corona product can be obtained as a corollary of Theorem2 by setting X=Vϒ1 and Y=Vϒ2.Corollary 1.
Letϒr,s be a graph and Sϒr′,s′ be its subdivision graph. Furthermore, it is supposed that ϒ1r1,s1 and ϒ2r2,s2 are two simple graphs. We assume X=Vϒ1 and Y=Vϒ2; then,(9)M1ϒS∘ϒ1,ϒ2=M1ϒ+rM1ϒ1+sM1ϒ2+r2+4sr1+rs1+ss2+rr1r1+1+s2+r22.Example 1.
LetCl denote a cycle on l vertices and Pn denote a path on n vertices. Then, by using the statement of Theorem 2, we get(10)M1ClS∘PmX,PnY=lr1′r1′+4+4lm+n+lr1′+r2′−8l+l2+r2′2+2l∑p∈Xdpmp+∑p∈Ydpnp,l≥3,m,n≥2.Theorem 3.
Letϒr,s be a graph and Sϒr′,s′ be its subdivision graph. Furthermore, it is supposed that ϒ1r1,s1 and ϒ2r2,s2 are two simple graphs. We assume X⊆Vϒ1 and Y⊆Vϒ2; then, the second Zagreb index of ϒS∘ϒ1X,ϒ2Y is given as(11)M2ϒS∘ϒ1X,ϒ2Y=∑pq∈ESϒpn∈Sϒqo∈ϒ2dϒqo+r2′dϒqo+2s′r1′+s′r1′r2′+rM2ϒ1+r∑pq∈Eϒ1p,q∈Xdϒ1p+dϒ1q+rEϒ1e=pqp,q∈X+r∑pq∈EΥ1p∉Xq∈Xdϒ1p+sM2ϒ2+s∑pq∈Eϒ2p,q∈Ydϒ2p+dϒ2q+sEϒ2e=pqp,q∈Y+s∑pq∈Eϒ2p∉Yq∈Ydϒ2p+2s+rr1′∑q∈XdΥ1q+2sr1′+rr1′2+s2+r2′∑q∈Ydϒ2q+sr2′2+r2′.Proof.
From the definition of the second Zagreb index, we have(12)M2ϒS∘ϒ1X,ϒ2Y=∑pq∈EϒS∘ϒ1X,ϒ2YdϒS∘ϒ1X,ϒ2Yp⋅dϒS∘ϒ1X,ϒ2Yq.
Now, applying Lemma1, we get(13)M2ϒS∘ϒ1X,ϒ2Y=∑pq∈ESϒpn∈Sϒqo∈Υdϒqo+r1′2+r2′+r∑pq∈Eϒ1p,q∈Xdϒ1p+1dϒ1q+1+r∑pq∈Eϒ1p∉Xq∈Xdϒ1pdϒ1q+1+r∑pq∈Eϒ1p∉Xq∉Xdϒ1pdϒ1q+s∑pq∈Eϒ2p,q∈Ydϒ2p+1dϒ2q+1+s∑pq∈Eϒ2p∉Yq∈Ydϒ2pdϒ2q+1+s∑pq∈Eϒ2p∉Yq∉Ydϒ2pdϒ2q+∑pqpo∈Sϒq∈Xdϒp+r1′dϒ1q+1+∑pqpn∈Sϒq∈Y2+r2′dϒ2q+1,=∑pq∈ESϒpn∈Sϒqo∈Υ2dϒqo+r2′dϒqo+2r1′+r1′r2′+r∑pq∈Eϒ1p,q∈Xdϒ1pdϒ1q+dϒ1p+dϒ1q+1+r∑pq∈Eϒ1p∉Xq∈Xdϒ1pdϒ1q+dϒ1p+r∑pq∈Eϒ1p∉Xq∉Xdϒ1pdϒ1q+s∑pq∈Eϒ2p,q∈Ydϒ2pdϒ2q+dϒ2p+dϒ2q+1+s∑pq∈Eϒ2p∉Yq∈Ydϒ2pdϒ2q+dΥ2p+s∑pq∈Eϒ2p∉Yq∉Ydϒ2pdϒ2q+∑pqpo∈Sϒq∈Xdϒpdϒ1q+dϒp+r1′dϒ1q+r1′+∑pqpn∈Sϒq∈Y2+r2′dϒ2q+1,=∑pq∈ESϒpn∈Sϒqo∈ϒ2dϒqo+r2′dϒqo+2r1′∑pq∈ESϒ1+r1′r2′∑pq∈ESϒ1+r∑pq∈Eϒ1p,q∈Xdϒ1pdϒ1q+r∑pq∈Eϒ1p,q∈Xdϒ1p+dϒ1q+r∑pq∈Eϒ1p,q∈X1+r∑pq∈Eϒ1p∉Xq∈Xdϒ1pdϒ1q+r∑pq∈Eϒ1p∉Xq∈Xdϒ1p+r∑pq∈Eϒ1p∉Xq∉Xdϒ1pdϒ1q+s∑pq∈Eϒ2p,q∈Ydϒ2pdϒ2q+s∑pq∈Eϒ2p,q∈Ydϒ2p+dϒ2q+s∑pq∈Eϒ2p,q∈Y1+s∑pq∈Eϒ2p∉Yq∈Ydϒ2pdϒ2q+s∑pq∈Eϒ2p∉Yq∈Ydϒ2p+s∑pq∈Eϒ2p∉Yq∉Ydϒ2pdϒ2q+∑p∈ϒdϒp⋅∑q∈Xdϒ1q+∑p∈ϒdϒp⋅∑q∈X1+r1′∑q∈Xdϒ1q⋅∑p∈ϒ1+r1′∑p∈ϒ1⋅∑q∈X1+2+r2′∑pn∈Sϒ1.∑q∈Ydϒ2q+2+r2′∑pn∈Sϒ1⋅∑q∈Y1,=∑pq∈ESϒpn∈Sϒqo∈Υ2dϒqo+r2′dϒqo+2r1′∑pq∈ESϒ1+r1′r2′∑pq∈ESϒ1+r∑pq∈Eϒ1p,q∈Xdϒ1pdϒ1q+∑pq∈Eϒ1p∉Xq∈XdΥ1pdϒ1q+∑pq∈Eϒ1p∉Xq∉Xdϒ1pdΥ1q+r∑pq∈Eϒ1p,q∈Xdϒ1p+dϒ1q+r∑pq∈Eϒ1p,q∈X1+r∑pq∈Eϒ1p∉Xq∈Xdϒ1p+s∑pq∈Eϒ2p,q∈Ydϒ2pdϒ2q+∑pq∈Eϒ2p∉Yq∈Ydϒ2pdϒ2q+∑pq∈Eϒ2p∉Yq∉Ydϒ2pdϒ2q+s∑pq∈Eϒ2p,q∈Ydϒ2p+dϒ2q+s∑pq∈Eϒ2p,q∈Y1+s∑pq∈Eϒ2p∉Yq∈Ydϒ2p+∑p∈ϒdϒp∑q∈Xdϒ1q+∑p∈ϒdϒp⋅∑q∈X1+r1′∑q∈Xdϒ1q⋅∑p∈ϒ1+r1′∑p∈ϒ1⋅∑q∈X1+2+r2′∑pn∈Sϒ1⋅∑q∈Ydϒ2q+2+r2′∑pn∈Sϒ1⋅∑q∈Y1,=∑pq∈ESϒpn∈Sϒqo∈Υ2dϒqo+r2′dϒqo+2r1′∑pq∈ESϒ1+r1′r2′∑pq∈ESϒ1+r∑pq∈Eϒ1dϒ1pdϒ1q+r∑pq∈Eϒ1p,q∈Xdϒ1p+dϒ1q+r∑pq∈Eϒ1p,q∈X1+r∑pq∈Eϒ1p∉Xq∈Xdϒ1p+s∑pq∈Eϒ2dϒ2pdϒ2q+s∑pq∈Eϒ2p,q∈Ydϒ2p+dϒ2q+s∑pq∈Eϒ2p,q∈Y1+s∑pq∈Eϒ2p∉Yq∈Ydϒ2p+∑p∈ϒdϒp⋅∑q∈Xdϒ1q+∑p∈ϒdϒp⋅∑q∈X1+r1′∑q∈XdΥ1q⋅∑p∈ϒ1+r1′∑p∈ϒ1⋅∑q∈X1+2+r2′∑pn∈Sϒ1⋅∑q∈Ydϒ2q+2+r2′∑pn∈Sϒ1⋅∑q∈Y1,=∑pq∈ESϒpn∈Sϒqo∈ϒ2dϒq+r2′dϒq+2s′r1′+s′r1′r2′+rM2ϒ1+r∑pq∈Eϒ1p,q∈Xdϒ1p+dϒ1q+rEϒ1e=pqp,q∈X+r∑pq∈Eϒ1p∉Xq∈Xdϒ1p+sM2ϒ2+s∑pq∈Eϒ2p,q∈Ydϒ2p+dϒ2q+sEϒ2e=pqp,q∈Y+s∑pq∈Eϒ2p∉Yq∈Ydϒ2p+2s+rr1′∑q∈Xdϒ1q+2sr1′+rr1′2+s2+r2′∑q∈Ydϒ2q+sr2′2+r2′,which is our required result.The classical subdivision double-corona product can be obtained as a corollary by settingX=Vϒ1 and Y=Vϒ2.Corollary 2.
Letϒr,s be a graph and Sϒr′,s′ be its subdivision graph. Furthermore, it is supposed that ϒ1r1,s1 and ϒ2r2,s2 are two simple graphs. We assume X=Vϒ1 and Y=Vϒ2; then,(14)M2ϒS∘ϒ1,ϒ2=rM1ϒ1+M2ϒ1+s1+sM1ϒ2+M2ϒ2+s2+2s12s+rr1+r12s′+s′r2+2s+rr1+2+r22ss2+sr2+∑pq∈ESϒpn∈Sϒqo∈ϒ2dϒqo+r2dϒqo.Example 2.
LetCl denote a cycle on l vertices and Pn denote a path on n vertices. Then, by using the statement of Theorem 3, we get(15)M2ClS∘PmX,PnY=∑pq∈ESClpn∈SClqo∈Cl2dClqo+r2′dClqo+2n−1r1′r2′+2+4lm+n−16l+l∑pq∈Epmp,q∈Xdpmp+dpmq+Epme=pqp,q∈X+∑pq∈Epmp∉Xq∈Xdpmp+l∑pq∈Epnp,q∈Ydpnp+dpnq+Epne=pqp,q∈Y+∑pq∈Epnp∉Yq∈Ydpnp+lr1′2+r1′+l2+r1′∑q∈Xdpmq+2+r2′∑q∈Ydpnq,l≥3,m,n≥2.Theorem 4.
Letϒr,s be a graph and Sϒr′,s′ be its subdivision graph. Furthermore, it is supposed that ϒ1r1,s1 and ϒ2r2,s2 are two simple graphs. We assume X=Vϒ1 and Y=Vϒ2; then, the first reformulated Zagreb index of ϒS∘ϒ1X,ϒ2Y is given as(16)EM1ϒS∘ϒ1X,ϒ2Y=∑pq∈ESϒpn∈Sϒqo∈Υdϒqo2+s′r1′+r2′2+2r1′+r2′∑pq∈Sϒpn∈SΥqo∈Υdϒqo+rHM1ϒ1+rEϒ1e=pqp∉Xq∈X−2r∑pq∈Eϒ1p∉Xq∈Xdϒ1p+dϒ1q+4rEϒ1e=pqp,q∉X−4r∑pq∈Eϒ1p∉Xq∉Xdϒ1p+dϒ1q+sHM1ϒ2+sEϒ2e=pqp∉Yq∈Y−2s∑pq∈Eϒ2p∉Yq∈Ydϒ2p+dϒ2q+4sEϒ2e=pqp,q∉Y−4s∑pq∈Eϒ2p∉Yq∉Ydϒ2p+dϒ2q+r1′M1ϒ+r∑q∈Xdϒ1q2+4s∑q∈Xdϒ1q+rr1′r1′−12+4sr1′r1′−1+2rr1′−1∑q∈Xdϒ1q+s∑q∈Ydϒ2q2+sr2′r2′+12+2sr2′+1∑q∈Ydϒ2q.Proof.
From the definition of the first reformulated Zagreb index, we have(17)EM1ϒS∘ϒ1X,ϒ2Y=∑pq∈EϒS∘ϒ1X,ϒ2YdϒS∘ϒ1X,ϒ2Yp+dϒS∘ϒ1X,ϒ2Yq−22.
Applying Lemma1, we get(18)EM1ϒS∘ϒ1X,ϒ2Y=∑pq∈ESϒpn∈Sϒqo∈ϒdϒqo+r1′+r2′2+r∑pq∈Eϒ1p,q∈Xdϒ1p+dϒ1q2+r∑pq∈Eϒ1p∉Xq∈Xdϒ1p+dϒ1q−12+r∑pq∈Eϒ1p∉Xq∉Xdϒ1p+dϒ1q−22+s∑pq∈Eϒ2p,q∈Ydϒ2p+dϒ2q2+s∑pq∈Eϒ2p∉Yq∈Ydϒ2p+dϒ2q−12+s∑pq∈Eϒ2p∉Yq∉Ydϒ2p+dϒ2q−22+∑pqpo∈Sϒq∈Xdϒp+dϒ1q+r1′−12+∑pqpn∈Sϒq∈Ydϒ2q+r2′+12,=∑pq∈ESϒpn∈Sϒqo∈ϒdϒqo2+r1′2+r2′2+2r1′r2′+2r1′+r2′dϒqo+r∑pq∈Eϒ1p,q∈Xdϒ1p+dϒ1q2+r∑pq∈Eϒ1p∉Xq∈Xdϒ1p+dϒ1q2+1−2dϒ1p+dϒ1q+r∑pq∈Eϒ1p∉Xq∉Xdϒ1p+dϒ1q2+4−4dϒ1p+dϒ1q+s∑pq∈Eϒ2p,q∈Ydϒ2p+dϒ2q2+s∑pq∈Eϒ2p∉Yq∈Ydϒ2p+dϒ2q2+1−2dϒ2p+dϒ2q+s∑pq∈Eϒ2p∉Yq∉Ydϒ2p+dϒ2q2+4−4dϒ2p+dϒ2q+∑pqpo∈Sϒq∈Xdϒp+dϒ1q2+r1′−12+2r1′−1dϒp+dϒ1q+∑pqpn∈Sϒq∈Ydϒ2q2+r2′+12+2r2′+1dϒ2q,=∑pq∈ESϒpn∈Sϒqo∈ϒdϒqo2+r1′2+r2′2+2r1′r2′∑pq∈ESϒ1+2r1′+r2′∑pq∈ESϒpn∈Sϒqo∈Υdϒqo+r∑pq∈Eϒ1p,q∈Xdϒ1p+dϒ1q2+r∑pq∈Eϒ1p∉Xq∈Xdϒ1p+dϒ1q2+r∑pq∈Eϒ1p∉Xq∈X1−2r∑pq∈Eϒ1p∉Xq∈Xdϒ1p+dϒ1q+r∑pq∈Eϒ1p∉Xq∉Xdϒ1p+dϒ1q2+4r∑pq∈Eϒ1p∉Xq∉X1−4r∑pq∈Eϒ1p∉Xq∉Xdϒ1p+dϒ1q+s∑pq∈Eϒ2p,q∈Ydϒ2p+dϒ2q2+s∑pq∈Eϒ2p∉Yq∈Ydϒ2p+dϒ2q2+s∑pq∈Eϒ2p∉Yq∈Y1−2s∑pq∈Eϒ2p∉Yq∈Ydϒ2p+dϒ2q+s∑pq∈Eϒ2p∉Yq∉Ydϒ2p+dϒ2q2+4s∑pq∈Eϒ2p∉Yq∉Y1−4s∑pq∈Eϒ2p∉Yq∉Ydϒ2p+dϒ2q+∑pqpo∈Sϒq∈Xdϒp+dϒ1q2+r1′−12∑pqpo∈Sϒq∈X1+2r1′−1∑pqpo∈Sϒq∈Xdϒp+dϒ1q+∑pqpn∈Sϒq∈Ydϒ2q2+r2′+12∑pqpn∈Sϒq∈Y1+2r2′+1∑pqpn∈Sϒq∈Ydϒ2q,=∑pq∈ESϒpn∈Sϒqo∈ϒdϒqo2+r1′2+r2′2+2r1′r2′∑pq∈ESϒ1+2r1′+r2′∑pq∈ESϒpn∈Sϒqo∈ϒdϒqo+r∑pq∈Eϒ1p,q∈Xdϒ1p+dϒ1q2+∑pq∈Eϒ1p∉Xq∈Xdϒ1p+dϒ1q2+∑pq∈Eϒ1p∉Xq∉Xdϒ1p+dϒ1q2+r∑pq∈Eϒ1p∉Xq∈X1−2r∑pq∈Eϒ1p∉Xq∈Xdϒ1p+dϒ1q+4r∑pq∈Eϒ1p∉Xq∉X1−4r∑pq∈Eϒ1p∉Xq∉Xdϒ1p+dϒ1q+s∑pq∈Eϒ2p,q∈Ydϒ2p+dϒ2q2+∑pq∈Eϒ2p∉Yq∈Ydϒ2p+dϒ2q2+∑pq∈Eϒ2p∉Yq∉Ydϒ2p+dϒ2q2+s∑pq∈Eϒ2p∉Yq∈Y1−2s∑pq∈Eϒ2p∉Yq∈Ydϒ2p+dϒ2q+4s∑pq∈Eϒ2p∉Yq∉Y1−4s∑pq∈Eϒ2p∉Yq∉Ydϒ2p+dϒ2q+∑pqpo∈Sϒq∈Xdϒp2+dϒ1q2+2dϒpdϒ1q+r1′−12∑pqpo∈Sϒq∈X1+2r1′−1∑pqpo∈Sϒq∈Xdϒp+dϒ1q+∑pqpn∈Sϒq∈Ydϒ2q2+r2′+12∑pqpn∈Sϒq∈Y1+2r2′+1∑pqpn∈Sϒq∈Ydϒ2q,=∑pq∈ESϒpn∈Sϒpo∈ϒdϒqo2+r1′2+r2′2+2r1′r2′∑pq∈ESϒ1+2r1′+r2′∑pq∈ESϒpn∈Sϒqo∈Υdϒqo+r∑pq∈Eϒ1dϒ1p+dϒ1q2+r∑pq∈Eϒ1p∉Xq∈X1−2r∑pq∈Eϒ1p∉Xq∈Xdϒ1p+dϒ1q+4r∑pq∈Eϒ1p∉Xq∉X1−4r∑pq∈Eϒ1p∉Xq∉Xdϒ1p+dϒ1q+s∑pq∈Eϒ2dϒ2p+dϒ2q2+s∑pq∈Eϒ2p∉Yq∈Y1−2s∑pq∈Eϒ2p∉Yq∈Ydϒ2p+dϒ2q+4s∑pq∈Eϒ2p∉Yq∉Y1−4s∑pq∈Eϒ2p∉Yq∉Ydϒ2p+dϒ2q+∑pqpo∈Sϒq∈Xdϒp2+∑pqpo∈Sϒq∈Xdϒ1q2+2∑pqpo∈Sϒq∈Xdϒpdϒ1q+r1′−12∑pqpo∈Sϒq∈X1+2r1′−1∑pqpo∈Sϒq∈Xdϒp+2r1′−1∑pqpo∈Sϒq∈Xdϒ1q+∑pqpn∈Sϒq∈Ydϒ2q2+r2′+12∑pqpn∈Sϒq∈Y1+2r2′+1∑pqpn∈Sϒq∈Ydϒ2q,=∑pq∈ESϒpn∈Sϒqo∈ϒdϒqo2+r1′2+r2′2+2r1′r2′∑pq∈ESϒ1+2r1′+r2′∑pq∈ESϒpn∈Sϒqo∈ϒdϒqo+r∑pq∈Eϒ1dϒ1p+dϒ1q2+r∑pq∈Eϒ1p∉Xq∈X1−2r∑pq∈Eϒ1p∉Xq∈Xdϒ1p+dϒ1q+4r∑pq∈Eϒ1p∉Xq∉X1−4r∑pq∈Eϒ1p∉Xq∉Xdϒ1p+dϒ1q+s∑pq∈EΥ2dϒ2p+dϒ2q2+s∑pq∈Eϒ2p∉Yq∈Y1−2s∑pq∈Eϒ2p∉Yq∈Ydϒ2p+dϒ2q+4s∑pq∈Eϒ2p∉Yq∉Y1−4s∑pq∈Eϒ2p∉Yq∉Ydϒ2p+dϒ2q+∑p∈ϒdϒp2∑q∈X.1+∑q∈Xdϒ1q2∑p∈ϒ⋅1+2∑p∈ϒdϒp∑q∈Xdϒ1q+r1′−12∑p∈ϒ1∑q∈X⋅1+2r1′−1∑p∈ϒdϒp∑q∈X⋅1+2r1′−1∑q∈Xdϒ1q∑p∈ϒ⋅1+∑q∈Ydϒ2q2∑pn∈Sϒ⋅1+r2′+12∑pn∈Sϒ⋅1∑q∈Y⋅1+2r2′+1∑q∈YdΥ2q∑pn∈Sϒ⋅1,=∑pq∈ESϒpn∈Sϒqo∈ϒdϒqo2+s′r1′+r2′2+2r1′+r2′∑pq∈ESϒpn∈Sϒqo∈Υdϒqo+rHM1ϒ1+rEϒ1e=pqp∉Xq∈X−2r∑pq∈Eϒ1p∉Xq∈Xdϒ1p+dϒ1q+4rEϒ1e=pqp,q∉X−4r∑pq∈Eϒ1p∉Xq∉Xdϒ1p+dϒ1q+sHM1ϒ2+sEϒ2e=pqp∉Yq∈Y−2s∑pq∈Eϒ2p∉Yq∈Ydϒ2p+dϒ2q+4sEϒ2e=pqp,q∉Y−4s∑pq∈Eϒ2p∉Yq∉Ydϒ2p+dϒ2q+r1′M1ϒ+r∑q∈Xdϒ1q2+4s∑q∈Xdϒ1q+rr1′r1′−12+4sr1′r1′−1+2rr1′−1∑q∈Xdϒ1q+s∑q∈Ydϒ2q2+sr2′r2′+12+2sr2′+1∑q∈Ydϒ2q,which is our required result.The classical subdivision double-corona product can be obtained as a corollary by settingX=Vϒ1 and Y=Vϒ2.Corollary 3.
Letϒr,s be a graph and Sϒr′,s′ be its subdivision graph. Furthermore, it is supposed that ϒ1r1,s1 and ϒ2r2,s2 are two simple graphs. We assume X=Vϒ1 and Y=Vϒ2; then,(19)EM1ϒS∘ϒ1,ϒ2=rHM1ϒ1+M1ϒ1+r1M1ϒ+rr1−12+sHM1ϒ2+M1ϒ2+8s1+r1−14sr1+4rs1+r2+1sr2r2+1+4ss2+∑pq∈ESϒpn∈Sϒqo∈ϒdΥqo2+r1+r2s′r1+r2+2∑pq∈ESϒpn∈Sϒqo∈ϒdϒqo.Example 3.
LetCl denote a cycle on l vertices and Pn denote a path on n vertices. Then, by using the statement of Theorem 4, we get(20)EM1ClS∘PmX,PnY=∑pq∈ESClpn∈SClqo∈CldClqo2+2n−1r1′+r2′2+2r1′+r2′∑pq∈ESClpn∈SClqo∈CldClqo+l16m−30+lEpme=pqp∉Xq∈X−2l∑pq∈Epmp∉Xq∈Xdpmp+dpmq+4lEpme=pqp,q∉X−4l∑pq∈Epmp∉Xq∉Xdpmp+dpmq+l16n−30+lEpne=pqp∉Yq∈Y−2l∑pq∈Epnp∉Yq∈Ydpnp+dpnq+4lEpne=pqp,q∉Y−4l∑pq∈Epnp∉Yq∉Ydpnp+dpnq+4r1′l+l∑q∈Xdpmq2+4l∑q∈Xdpmq+lr1′r1′−12+4lr1′r1′−1+2lr1′−1∑q∈Xdpmq+l∑q∈Ydpnq2+lr2′r2′+12+2lr2′+1∑q∈Ydpnq,l≥3,m,n≥2.Theorem 5.
Letϒr,s be a graph and Sϒr′,s′ be its subdivision graph. Furthermore, it is supposed that ϒ1r1,s1 and ϒ2r2,s2 are two simple graphs. We assume X=Vϒ1 and Y=Vϒ2; then, the forgotten topological index of ϒS∘ϒ1X,ϒ2Y is given as(21)FϒS∘ϒ1X,ϒ2Y=Fϒ+3r1′M1ϒ+6sr1′2+rr1′3+rFϒ1+3r∑p∈Xdϒ1p2+3r∑p∈Xdϒ1p+rr1′+sFϒ2+3s∑p∈Ydϒ2p2+3s∑p∈Ydϒ2p+sr2′+s2+r2′3.Proof.
From the definition of the forgotten topological index, we have(22)FϒS∘ϒ1X,ϒ2Y=∑p∈VϒS∘ϒ1X,ϒ2YdϒS∘ϒ1X,ϒ2Yp3.
Applying Lemma1, we get(23)FϒS∘ϒ1X,ϒ2Y=∑p∈Vϒdϒp+r1′3+r∑p∈Xdϒ1p+13+r∑p∉Xdϒ1p2+s∑p∈Ydϒ2p+13+s∑p∉Ydϒ2p3+∑p∈VSϒ2+r2′3,=∑p∈Vϒdϒp3+3r1′dϒp2+3r1′2dϒp+r1′3+r∑p∈Xdϒ1p3+3dϒ1p2+3dϒ1p+1+r∑p∉Xdϒ1p3+s∑p∈Ydϒ2p3+3dϒ2p2+3dϒ2p+1+s∑p∉Ydϒ2p3+∑p∈VSϒ2+r2′3,=∑p∈Vϒdϒp3+3r1′∑p∈Vϒdϒp2+3r1′2∑p∈Vϒdϒp+r1′3∑p∈Vϒ1+r∑p∈Xdϒ1p3+3r∑p∈Xdϒ1p2+3r∑p∈Xdϒ1p+r∑p∈X1+r∑p∉Xdϒ1p3+s∑p∈Ydϒ2p3+3s∑p∈Ydϒ2p2+3s∑p∈Ydϒ2p+s∑p∈Y1+s∑p∉Ydϒ2p3+∑p∈VSϒ2+r2′3,=∑p∈Vϒdϒp3+3r1′∑p∈Vϒdϒp2+3r1′2∑p∈Vϒdϒp+r1′3∑p∈Vϒ1+r∑p∈Xdϒ1p3+∑p∉Xdϒ1p3+3r∑p∈Xdϒ1p2+3r∑p∈Xdϒ1p+r∑p∈X1+s∑p∈Ydϒ2p3+∑p∉Ydϒ2p3+3s∑p∈Ydϒ2p2+3s∑p∈Ydϒ2p+s∑p∈Y1+∑p∈VSϒ2+r2′3,=∑p∈Vϒdϒp3+3r1′∑p∈Vϒdϒp2+3r1′2∑p∈VϒdΥp+r1′3∑p∈Vϒ1+r∑p∈Vϒ1dϒ1p3+3r∑p∈Xdϒ1p2+3r∑p∈Xdϒ1p+r∑p∈X1+s∑p∈Vϒ2dϒ2p3+3s∑p∈Ydϒ2p2+3s∑p∈Ydϒ2p+s∑p∈Y1+2+r2′3∑p∈VSϒ1=Fϒ+3r1′M1ϒ+6sr1′2+rr1′3+rFϒ1+3r∑p∈Xdϒ1p2+3r∑p∈Xdϒ1p+rr1′+sFϒ2+3s∑p∈Ydϒ2p2+3s∑p∈Ydϒ2p+sr2′+s2+r2′3,which is our required result.The classical subdivision double-corona product can be obtained as a corollary by settingX=Vϒ1 and Y=Vϒ2.Corollary 4.
Letϒr,s be a graph and Sϒr′,s′ be its subdivision graph. Furthermore, it is supposed that ϒ1r1,s1 and ϒ2r2,s2 are two simple graphs. We assume X=Vϒ1 and Y=Vϒ2; then,(24)FϒS∘ϒ1,ϒ2=Fϒ+rFϒ1+sFϒ2+3r1M1ϒ+3rM1ϒ1+3sM1ϒ2+rr1r12+1+sr2+r2+23+6ss2+sr12+s1r.Example 4.
LetCl denote a cycle on l vertices and Pn denote a path on n vertices. Then, by using the statement of Theorem 5, we get(25)FClS∘PmX,PnY=8lm+n+lr2′−20+lr1′r1′2+6r1′+13+l2+r2′3+3l∑p∈Xdpmp2+∑p∈Xdpmp+3l∑p∈Ydpnp2+∑p∈Ydpnp,l≥3,m,n≥2.


## 3. Conclusions

In this paper, we have given explicit expressions to compute the first Zagreb, second Zagreb, first reformulated Zagreb index, and forgotten topological index of the generalized subdivision double-corona product of graphs. The obtained results are the generalization of the results obtained by Jamil and Hameed. Moreover, in each case, an example is provided to compute the generalized subdivision double-corona product of cycle and path graphs.



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*Source: 3119739-2022-01-18.xml*