# Two Complex Graph Operations and their Exact Formulations on Topological Properties

Authors: Shehla Hameed; Muhammad Kamran Jamil; Muhammad Waheed; Muhammad Azeem; Senesie Swaray
Journal: Complexity
 (2022)

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## Abstract

Graph operations are utilized for developing complicated graph structures from basic graphs, and these basic graphs can help to understand the properties of complex networks. While on the other side, the topological descriptor is known as a numeric value that is associated with the graph of a network. It has enormous practical applications in chemistry and other fields of science. This particular work in this draft is the extended work and investigated the first, second, first multiplicative, first reformulated Zagreb indices, and the forgotten index of subdivision double corona and subdivision double neighborhood corona products.


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## Body

## 1. Introduction

A topological index is a number associated with a graph of some network. With the help of this number, we can describe some properties of the network. Especially in organic chemistry, topological indices are used to predict some physical, chemical, or biological properties of organic compounds. Topological indices are key topics in the study of quantitative structural properties of a chemical network [1–5].For a graphλ=Vλ,Eλ, the vertex and edge sets are denoted by Vλ and Eλ. The number of elements in Vλ and Eλ is called the order n and the size m, respectively, of λ. A graph of order n and size m is denoted by λn,m. The set of vertices adjacent to the vertex v∈Vλ is called the neighborhood set of v and the number of elements in the neighborhood set the degree of v in λ is denoted by dλv [6–10].The study of the topological index started in 1947, and after that, hundreds of topological indices have been presented depending on the nature of applications for different chemical compounds. In 1972, the researchers in [11] introduced the first and second Zagreb indices. For a graph λ, these topological indices are defined as(1)M1λ=∑v∈Vλdλv2=∑uv∈Eλdλu+dλv,M2λ=∑uv∈Eλdλudλv.The researchers of [12, 13], introduced multiplicative variants of ordinary Zagreb indices. These topological indices are used to study molecular chirality, complexity, heterosystems, and Ze-isomerism. The first and second multiplicative Zagreb indices are defined as(2)∏1λ=∏v∈Vλdλv2,∏2λ=∏uv∈Eλdλudλv.In 2015, researchers in [14, 15] suggested a forgotten topological index that is comparable to the first Zagreb index in its applications. The forgotten topological index is also known as F-index, and it is defined as(3)Fλ=∑v∈Vλdλv3=∑uv∈Eλdλu2+dλv2.In 2004, Milicevic et al. [16] proposed reformulated Zagreb indices using edge-degrees rather than vertex-degrees. Mathematically, it is expressed as(4)EM1λ=∑e∈Eλde2wherede=du+dv−2.Forde=du+dv, the above expression is known as the first hyper Zagreb index HM1λ.Complex network structures or large molecular structures can be constructed by applying some graph operations on simple graphs. Furthermore, these simple graphs can help to describe some properties of these structures. For example, the Cartesian product provides a significant model for connecting computers [17, 18].For graphsλ1n1,m1 and λ2n2,m2, the corona product λ1°λ2 is obtained by taking one copy of λ1, n1 copies of λ2, and joining jth vertex of λ1 to every vertex Jth copy of λ2 [19]. A special graph obtained by attaching a vertex in the each edge of λ is called the subdivision graph of λ and is symbolized by λs [20].Letλn,m, λ1n1,m1, and λ2n2,m2 be three graphs. The operation known as subdivision double corona product of λ, λ1, and λ2 and symbolized by λs°λ1,λ2, and is attained by making single copy of λs,n copies of λ1,m copies of λ2, and after that, by attaching the ith old vertex, uo, of λs to each vertex of the ith copy of λ1 and jth new vertex uN of λs to each vertex of the jth copy of λ2 [21]. An illustration of subdivision double corona product is shown in Figure 1.Figure 1
Subdivision double corona productC4s°P3,P2.For the above three graphs, the subdivision double neighborhood corona product,λs⋅λ1,λ2, is the graph attained by making single copy of λs,n copies of λ1,m copies of λ2 and after that, by attaching the neighborhood vertices of the ith old vertex uo of λs to every vertex of the ith duplicate of λ1 and joining the neighborhood vertices of the jth new vertex uN of λs to every vertex of the jth duplicate of λ2 [21]. Figure 2 explains the notation of λs⋅λ1,λ2.Figure 2
Subdivision double neighborhood corona productC4s⋅P3,P2.In [22], the authors investigated the first and second Zagreb indices of the Cartesian, composition, join, disjunction, and symmetric difference graph operations. The author in [23] computed the forgotten topological index of different corona products of graphs and the author in [24] gave the exact expressions of Zagreb indices of the generalized hierarchical product of graphs. For more discussion and results, we refer to [25, 26]. There are some new and recent topics related to this study is found, one can see [27–31].The Laplacian spectrum of double neighborhood corona graphs are found in the literature of [21], and the main results are presented.Theorem 1.
Letλ be a t-regular graph on n vertices, m edges, λ1 and λ2 be any two graphs on n1 and n2 vertices, respectively. Then, the Laplacian spectrum of λs°λ1.λ2 comprises(i)
η4−n1+n2+t+4η4+n1+1n2+3+2t+1+n2t+ηjλη2−tn2+1+2n1+ηjλ+1η+ηjλ=0, for 1≤j≤n;(ii)
n2+3±n2+32−8/2 repeated m−n times each;(iii)
ηjλ1+1 repeated n times, for 2≤j≤n1;(iv)
ηjλ2+1 repeated m times, for 2≤j≤n2.Theorem 2 (see [21]).
Letλ be a t-regular graph on n vertices, m edges, λ1 and λ2 be any two graphs on n1 and n2 vertices, respectively. Then, the Laplacian spectrum of λs⋅λ1,λ2 comprises(i)
all the roots of the equation(5)η−n1+n2+t+4+n2+4n1+t+1−2r−ηjλn1+n2−2t−ηjλη2−n1+n−2+t+2−2t−ηjλn1n1+t+1+n2n2+3+2n1+tn2+2−2tηjλ+n1+tn2+2−2t−ηjλ+2t−ηjλn1n2−n1−n2=0,for1≥j≥n.(ii)
n2+2 repeated m−n times;(iii)
1 repeatedm−n times;(iv)
ηiλ1 + 1 repeated n times, for 2≤i≤n1;(v)
ηiλ2 + 1 repeated n times, for 2≤i≤n2.In this paper, we extend the work and investigated some degree-based topological indices of these graph operations.


## 2. Main Results

The current section contained the main results which include the formulation of some degree-based topological indices such as first and second Zagreb, first multiplicative Zagreb, first reformulated Zagreb, the forgotten indices of subdivision double corona product, and subdivision double neighborhood corona product of graphs.Following is the famous relationship between arithmetic and geometric means.Lemma 1.
(AM-GM Inequality). Leta1,a2,…,an be non-negative numbers. Then,(6)a1+a2+⋯+an2≥a1a2…an1/n,and the equality holds if and only if a1=a2=…=an are equal.Next to lemmas are the direct results from the definitions of the subdivision double corona product and subdivision neighborhood corona product of graphs.Lemma 2.
Letλ, λ1, and λ2 are three graphs having order n, n1, and n2, respectively. Then, the degree behavior of the vertices in subdivision double corona product is given as(7)dλs°λ1,λ2v=dλv+n1ifv∈Vλ;dλ1v+1ifv∈Vλ1;dλ2v+1ifv∈Vλ2;n2+2ifv∈Vλs.Lemma 3.
Letλ, λ1, and λ2 are graphs having order n, n1, and n2, respectively, then the degrees of the vertices in subdivision double neighborhood corona product is(8)dλs•λ1,λ2v=dλv+2n2ifv∈Vλ;dλ1v+2ifv∈Vλ1;dλ2v+2ifv∈Vλ2;21+n1ifv∈Vλs.Following is our first main result, which gives the first Zagreb index of the subdivision double corona product in terms of the first Zagreb indices of basic graphs, their orders, and sizes.Theorem 3.
Letλn,m, λ1n1,m1, and λ2n2,m2 be the simple connected graphs. Then, the first Zagreb index of the subdivision double corona product, λs°λ1,λ2, is given as(9)M1λs°λ1,λ2=M1λ+4mn1+nn12+nM1λ1+4m1n+nn1+mM1λ2+4m2m+mn2+m2+n22.Proof.
From the concept of topological descriptor named the first Zagreb index, we have got(10)M1λs°λ1,λ2=∑v∈Vλs°λ1,λ2dλs°λ1,λ2v2.
Now, we apply Lemma2,(11)M1λs°λ1,λ2=∑v∈Vλdλv+n12+n∑v∈Vλ1dλ1v+12+m∑v∈Vλ2dλ2v+12+∑v∈Vλs2+n22=∑v∈Vλdλv2+2n1∑v∈Vλdλv+n12∑v∈Vλ1+n∑v∈Vλ1dλ1v2+2∑v∈Vλ1dλ1v+∑v∈Vλ11m∑v∈Vλ2dλ2v2+2∑v∈Vλ2dλ2v+∑v∈Vλ21+2+n22∑v∈Vλs1=M1λ+4mn1+nn12+nM1λ1+4m1n+nn1+mM1λ2+4m2m+mn2+m2+n22.
Hence, the required expression.
The next results put a bound on the first multiplicative Zagreb index for the subdivision double corona product.Theorem 4.
Letλn,m, λ1n1,m1, and λ2n2,m2 be the simple connected graphs. Then, the first multiplicative Zagreb index of the subdivision double corona product λs°λ1,λ2 is given as(12)∏1λs°λ1,λ2≤M1λ+4n1m+nn12nn⋅M1λ1+4m1+n1n1nn1⋅M1λ2+4m2+n2n2mn2⋅2+n22m.Proof.
From using the concept of the first multiplicative Zagreb index and Lemma2, we have got(13)∏1λs°λ1,λ2=∏v∈Vλs°λ1,λ2dλs°λ1,λ2v2=∏v∈Vλdλv+n12×∏v∈Vλ1dλ1v+12n×∏v∈Vλ2dλ2v+12m×∏v∈Vλs2+n22=∏v∈Vλdλv2+2dλvn1+n12×∏v∈Vλ1dλ1v2+2dλ1v+1n×∏v∈Vλ2dλ2v2+2dλ2v+1m×∏v∈Vλs2+n22≤∑v∈Vλdλv2+2dλvn1+n12nn×∑v∈Vλ1dλ1v2+2dλ1v+1n1nn1×∑v∈Vλ2dλ2v2+2dλ2v+1n2mn2×∑v∈Vλs2+n22mm≤∑v∈Vλdλv2+2n1∑v∈Vλdλv+n12∑v∈Vλ1nn×∑v∈Vλ1dλ1v2+2∑v∈Vλ1dλ1v+∑v∈Vλ11n1nn1×∑v∈Vλ2dλ2v2+2∑v∈Vλ2dλ2v+∑v∈Vλ21n2mn2×2+n22∑v∈Vλs1mm=M1λ+4n1m+nn12nn×M1λ1+4m1+n1n1nn1×M1λ2+4m2+n2n2mn2×2+n22m.
The inequality is due to the Lemma1. Equality in the last expression holds if and only if λ, λ1, and λ2 are regular graphs.Theorem 5.
Letλn,m, λ1n1,m1, and λ2n2,m2 are three simple graphs and λsn′,m′ be the subdivided graph of λ. Then, the second Zagreb index of subdivision double corona product λs°λ1,λ2 is given as(14)M2λs°λ1,λ2=nM1λ1+mM1λ2+nM2λ1+mM2λ2+4mm1+m2+2n1+n1n2m′+mm2+n13m+2m1n+nn1+mn22+n2+2m2+∑uv∈Eλs2dλv+n2dλv.Proof.
From using the concept of the second Zagreb index and Lemma2, we have got(15)M2λs°λ1,λ2=∑uv∈Eλs°λ1,λ2dλs°λ1,λ2u⋅dλs°λ1,λ2v.=∑uv∈Eλsdλ+n12+n2+n∑uv∈Eλ1dλ1u+1dλ1v+1+m∑uv∈Eλ2dλ2u+1dλ2v+1+∑uo∈Vλsv∈Vλ1dλu+n1dλ1v+1+∑uN∈Vλsv∈Vλ22+n2dλ2v+1=∑uv∈Eλs2dλu+n2dλu+2n1+n1n2+n∑uv∈Eλ1dλ1udλ1v+dλ1u+dλ1v+1+m∑uv∈Eλ2dλ2udλ2v+dλ2u+dλ2v+1+∑uo∈Vλsv∈Vλ1dλudλ1v+dλu+n1dλ1v+n1+∑uN∈Vλsv∈Vλ22dλ2v+2+n2dλ2v+n2=∑uv∈Eλs2dλu+n2dλu+∑uv∈Eλs2n1+n1n2+n∑uv∈Eλ1dλ1udλ1v+∑uv∈Eλ1dλ1u+dλ1v+∑uv∈Eλ11+m∑uv∈Eλ2dλ2udλ2v+∑uv∈Eλ2dλ2u+dλ2v+∑uv∈Eλ21++∑uo∈Vλsv∈Vλ1dλudλ1v+∑uo∈Vλsv∈Vλ1dλu+n1dλ1v+n1∑uo∈Vλsv∈Vλ11+∑uN∈Vλsv∈Vλ22dλ2v+n2dλ2v+∑uN∈Vλsv∈Vλ22+n2∑uv∈Eλs2dλu+n2dλu+2n1+n1n2Eλs+nM2λ1+nM1λ1+nEλ1+mM2λ2+mM1λ2+mEλ2+∑uo∈Vλsdλu∑v∈Vλ1dλ1v+∑uo∈Vλsdλu∑v∈Vλ11+n1∑uo∈Vλs1∑v∈Vλ1dλ1v+n1∑uo∈Vλs1∑v∈Vλ11+2∑uN∈Vλs1∑v∈Vλ2dλ2v+n2∑uN∈Vλs1∑v∈Vλ2dλ2v+2+n2∑uN∈Vλs1∑v∈Vλ21=∑uv∈Eλs2dλu+n2dλu+2n1+n1n2m′+nM2λ1+nM1λ1+nm1+mM2λ2+mM1λ2+mm2+4mm1+2mn1+2m1nn1+nn12+4mm2+2mm2n2+mn22+n2.
After some simplification, we can get the required result.
The next result is about the first reformulated Zagreb index of the subdivision double corona product of graphs.Theorem 6.
Letλn,m, λ1n1,m1, and λ2n2,m2 be the simple graphs and λsn′,m′ is the subdivision of the graph λ. Then, the first reformulated Zagreb index of the subdivision double corona product λs°λ1,λ2 is given as(16)EM1λs°λ1,λ2=n1M1λ+nM1λ1+mM1λ2+nHM1λ1+mHM1λ2+n1+n22m′+8mm1+nn1n1−12+2n1−12mn1+2nm1+1+n22mn2+4mm21+n2+∑uv∈Eλsdλu2+2n1+n2dλu.Proof.
Using the Lemma2 and the concept of the first reformulated Zagreb index, we have got(17)EM1λs°λ1,λ2=∑uv∈Eλs°λ1,λ2dλs°λ1,λ2u+dλs°λ1,λ2v−22=∑uv∈Eλsdλu+n1+2+n2−22+n∑uv∈Eλ1dλ1u+1+dλ1v+1−22+m∑uv∈Eλ2dλ2u+1+dλ2v+1−22+∑uo∈Vλsv∈Vλ1dλu+n1+dλ1v+1−22+∑uN∈Vλsv∈Vλ22+n2+dλ2v+1−22=∑uv∈Eλsdλu+n1+n22+n∑uv∈Eλ1dλ1u+dλ1v2+m∑uv∈Eλ2dλ2u+dλ2v2+∑uo∈Vλsv∈Vλ1dλu+dλ1v+n1−12+∑uN∈Vλsv∈Vλ2dλ2v+1+n22=∑uv∈Eλsdλu2+n1+n22+2dλun1+n2+n∑uv∈Eλ1dλ1u+dλ1v2+m∑uv∈Eλ2dλ2u+dλ2v2+∑uo∈Vλsv∈Vλ1dλu+dλ1v2+n1−12+2dλu+dλ1vn1−1+∑uN∈Vλsv∈Vλ2dλ2v2+1+n22+2dλ2v1+n2=∑uv∈Eλsdλu2+2n1+n2∑uv∈Eλsdλu+∑uv∈Eλsn1+n22+nHM1λ1+mHM1λ2+∑uo∈Vλsv∈Vλ1dλu+dλ1v2+n1−12∑uo∈Vλsv∈Vλ11+2n1−1∑uo∈Vλsv∈Vλ1dλu+dλ1v+∑uN∈Vλsv∈Vλ2dλ2v2+1+n22∑uN∈Vλsv∈Vλ21+21+n2∑uN∈Vλsv∈Vλ2dλ2v=∑uv∈Eλsdλu2+2n1+n2dλu+n1+n22m′+nHM1λ1+mHM1λ2+∑uo∈Vλsv∈Vλ1dλu2+∑uo∈Vλsv∈Vλ1dλ1v2+2∑uo∈Vλsv∈Vλ1dλudλ1v+nn1n1−12+2n1−1∑uo∈Vλsv∈Vλ1dλu+dλ1v+∑uN∈Vλsv∈Vλ2dλ2v2+mn21+n22+21+n2∑uN∈Vλsv∈Vλ2dλ2v=∑uv∈Eλsdλu2+2n1+n2dλu+n1+n22m′+nHM1λ1+mHM1λ2+∑uo∈Vλsdλu2∑v∈Vλ11+∑uo∈Vλs1∑v∈Vλ1dλ1v2+2∑uo∈Eλsdλu∑v∈Vλ1dλ1v+nn1n1−12+2n1−1∑uo∈Vλsdλu∑v∈Vλ11+∑uo∈Vλs1∑v∈Vλ1dλ1v+∑uN∈Vλs1∑v∈Vλ2dλ2v2+mn21+n22+21+n2∑uN∈Vλs1∑v∈Vλ2dλ2v=∑uv∈Eλsdλu2+2n1+n2dλu+n1+n22m′+nHM1λ1+mHM1λ2+n1M1λ+nM1λ1+2⋅2m⋅2m1+n1−12⋅nn1+2n1−12mn1+2m1n+mM1λ2+1+n22mn2+21+n2.2m2m=∑uv∈Eλsdλu2+2n1+n2dλu+n1+n22m′+nHM1λ1+mHM1λ2+n1M1λ+nM1λ1+8mm1+nn1n1−12+4n1−1mn1+nm1+mM1λ2+mn21+n22+4mm21+n2.
Hence, the proof is done.Theorem 7.
For graphsλn,m, λ1n1,m1, and λ2n2,m2, then, the first Zagreb index of the subdivision double neighborhood corona product λs⋅λ1,λ2 is given as(18)M1λs⋅λ1,λ2=M1λ+nM1λ1+mM1λ2+8mn2+4n22n+8nm1+4nn1+8mm2+4mn2+m2+2n12.Proof.
From the concept of the first Zagreb index we have(19)M1λs⋅λ1,λ2=∑v∈Vλs⋅λ1,λ2dλs⋅λ1,λ2v2.
Now, we apply Lemma3,(20)=∑v∈Vλdλv+2n22+n∑v∈Vλ1dλ1v+22+m∑v∈Vλ2dλ2v+22+∑v∈Vλs2+2n12=∑v∈Vλdλv2+2dλv2n2+2n22+n∑v∈Vλ1dλ1v2+22dλ1v+22+m∑v∈Vλ2dλ2v2+2dλ2v2+22+∑v∈Vλs2+2n12=∑v∈Vλdλv2+4n2∑v∈Vλdλv+4n22∑v∈Vλ1+n∑v∈Vλ1dλ1v2+4∑v∈Vλ1dλ1v+4∑v∈Vλ11+m∑v∈Vλ2dλ2v2+4∑v∈Vλ2dλ2v+4∑v∈Vλ21+2+2n12∑v∈Vλs1=M1λ+8mn2+4n22n+nM1λ1+8nm1+4nn1+mM1λ2+8mm2+4mn2+m2+2n12.
Hence, the proof is done.Theorem 8.
Letλn,m, λsn′,m′, λ1n1,m1, and λ2n2,m2 be the simple connected graphs. Then, the second Zagreb index of subdivision double neighborhood corona product λs⋅λ1,λ2 is given as(21)M2λs⋅λ1,λ2=2∑uv∈Eλsdλu+n1dλu+4n2m′+4n1n2m′+nM2λ1+2nM1λ1+4nm1+mM2λ2+2mM1λ2+12mm2+8mn2+8nn2m2+8nn22+8mm1+8mm1n1+8mn1+8mn12.Proof.
From the concept of the second Zagreb index, we have(22)M2λs⋅λ1,λ2=∑uv∈Eλs⋅λ1,λ2dλs⋅λ1,λ2u.dλs⋅λ1,λ2v.
Now, we apply Lemma3,(23)=∑uv∈Eλsdλu+2n22+2n1+n∑uv∈Eλ1dλ1u+2dλ1v+2+m∑uv∈Eλ2dλ2u+2dλ2v+2+2∑uo∈Vλsv∈Vλ2dλu+2n2dλ2v+2+2∑uN∈Vλsv∈Vλ12+2n1dλ1v+2=∑uv∈Eλs2dλu+2n1dλu+4n2+4n1n2+n∑uv∈Eλ1dλ1udλ1v+2dλ1u+2dλ1v+4+m∑uv∈Eλ2dλ2udλ2v+2dλ2u+2dλ2v+4+2∑uo∈Vλsv∈Vλ2dλudλ2v+2dλu+2n2dλ2v+4n2+2∑uN∈Vλsv∈Vλ12dλ1v+4+2n1dλ1v+4n1=2∑uv∈Eλsdλu+n1dλu+4n2∑uv∈Eλs1+4n1n2∑uv∈Eλs1+n∑uv∈Eλ1dλ1udλ1v+2∑uvıEλ1dλ1u+dλ1v+4∑uv∈Eλ11+m∑uv∈Eλ2dλ2udλ2v+2∑uv∈Eλ2dλ2u+dλ2v+4∑uv∈Eλ21+2∑uo∈Vλsv∈Vλ2dλudλ2v+∑uo∈Vλsv∈Vλ22dλu+2n2dλ2v+4n2∑uo∈Vλsv∈Vλ21+2∑uN∈Vλsv∈Vλ12dλ1v+2n1dλ1v+4∑uN∈Vλsv∈Vλ11+4n1∑uN∈Vλsv∈Vλ11=2∑uv∈Eλsdλu+n1dλu+4n2m′+4n1n2m′+nM2λ1+2M1λ1+4m1+mM2λ2+2M1λ2+4m2+2∑uo∈Vλsdλu∑v∈Vλ2dλ2v+2∑uo∈Vλsdλu∑v∈Vλ21+2n2∑uo∈Vλs1∑v∈Vλ2dλ2v+4n2∑uo∈Vλs1∑v∈Vλ21+22∑uN∈Vλs1∑v∈Vλ1dλ1v+2n1∑uN∈Vλs1∑v∈Vλ1dλ1v+4∑uN∈Vλs1∑v∈Vλ11+4n1∑uN∈Vλs1∑v∈Vλ11=2∑uv∈Eλsdλu+n1dλu+4n2m′+4n1n2m′+nM2λ1+2M1λ1+4m1+mM2λ2+2M1λ2+4m2+22m⋅2m2+2⋅2m⋅n2+2n2n⋅2m2+4n2n⋅n2+22m⋅2m1+2n1m⋅2m1+4mn1+4mn12=2∑uv∈Eλsdλu+n1dλu+4n2m′+4n1n2m′+nM2λ1+2M1λ1+4m1+mM2λ2+2M1λ2+4m2+24mm2+4mn2+4nn2m2+4nn22+24mm1+4mm1n1+4mn1+4mn12=2∑uv∈Eλsdλu+n1dλu+4n2m′+4n1n2m′+nM2λ1+2nM1λ1+4nm1+mM2λ2+2mM1λ2+12mm2+8mn2+8nn2m2+8nn22+8mm1+8mm1n1+8mn1+8mn12.
Hence, the proof is done.Theorem 9.
Letλn,m,λ1n1,m1, and λ2n2,m2 be the simple connected graphs. Then, the forgotten topological index of subdivision double neighborhood corona product λs⋅λ1,λ2 is given as(24)Fλs⋅λ1,λ2=Fλ+24mn22+6n2M1λ+8nn23+nFλ1+24nm1+6nM1λ1+8nn1+mFλ2+24mm2+6mM1λ2+8mn2+m2+2n13.Proof.
From the concept of the forgotten index, we have(25)Fλs⋅λ1,λ2=∑v∈Vλs•λ1,λ2dλs⋅λ1,λ2v3.
Now, we apply Lemma3,(26)=∑v∈Vλdλv+2n23+n∑v∈Vλ1dλ1v+23+m∑v∈Vλ2dλ2v+23+∑v∈Vλs2+2n13=∑v∈Vλdλv3+3dλv2n22+3dλv22n2+2n23+n∑v∈Vλ1dλ1v3+3dλ1v22+3dλ1v2⋅2+23+m∑v∈Vλ2dλ2v3+3dλ2v22+3dλ2v2⋅2+23+∑v∈Vλs2+2n13=∑v∈Vλdλv3+12n22∑v∈Vλdλv+6n2∑v∈Vλdλv2+8n23∑v∈Vλ1+n∑v∈Vλ1dλ1v3+12n∑v∈Vλ1dλ1v+6n∑v∈Vλ1dλ1v2+8n∑v∈Vλ11+m∑v∈Vλ2dλ2v3+12m∑v∈Vλ2dλ2v+6m∑v∈Vλ2dλ2v2+8m∑v∈Vλ21+2+2n13∑v∈Vλs1=Fλ+24mn22+6n2M1λ+8nn23+nFλ1+24nm1+6nM1λ1+8nn1+mFλ2+24mm2+6mM1λ2+8mn2+m2+2n13.
Hence, the proof is done.Theorem 10.
Letλn,m, λ1n1,m1, and λ2n2,m2 be the simple connected graphs. Then, the first multiplicative Zagreb index of subdivision double neighborhood corona product λs⋅λ1,λ2 is given as(27)∏1λs⋅λ1,λ2≤M1λ+8n2m+4nn22nn×M1λ1+8m1+4n1n1nn1×M1λ2+8m2+4n2n2mn2×2+2n12m.Proof.
From the concept of the first multiplicative Zagreb index, we have(28)∏1λs⋅λ1,λ2=∏v∈Vλs⋅λ1,λ2dλs⋅λ1,λ2v2.
Now, we apply Lemma3,(29)=∏v∈Vλdλv+2n22×∏v∈Vλ1dλ1v+22n×∏v∈Vλ2dλ2v+22m×∏v∈Vλs2+2n12=∏v∈Vλdλv2+2dλv2n2+2n22×∏v∈Vλ1dλ1v2+2dλ1v×2+22n×∏v∈Vλ2dλ2v2+2dλ2v×2+22m×∏v∈Vλs2+2n12.
By Lemma1,(30)≤∑v∈Vλdλv2+4n2dλv+4n22nn×∑v∈Vλ1dλ1v2+4dλ1v+4n1nn1×∑v∈Vλ2dλ2v2+4dλ2v+4n2mn2×∑v∈Vλs2+2n12mm≤∑v∈Vλdλv2+4n2∑v∈Vλdλv+4n22∑v∈Vλ1nn×∑v∈Vλ1dλ1v2+4∑v∈Vλ1dλ1v+4∑v∈Vλ11n1nn1×∑v∈Vλ2dλ2v2+4∑v∈Vλ2dλ2v+4∑v∈Vλ21n2mn2×2+2n12∑v∈Vλs1mm=M1λ+8n2m+4nn22nn×M1λ1+8m1+4n1n1nn1×M1λ2+8m2+4n2n2mn2×2+2n12m.
Hence, equality holds in 13 iffλ, λ1, and λ2 are regular graphs.
Hence, the proof is done.Theorem 11.
Letλn,m, λsn′,m′, λ1n1,m1, and λ2n2,m2 be the simple connected graphs. Then, the first reformulated Zagreb index of subdivision double neighborhood corona product λs⋅λ1,λ2 is given as(31)EM1λs⋅λ1,λ2=∑uv∈Eλsdλu2+4n1+n2dλu+4n1+n22m′+nHM1λ1+4nm1+4nM1λ1+mHM1λ2+4mm2+4mM1λ2+2n2M1λ+2nM1λ2+6mm2+8nn23+16mn22+16nn2m2+2mM1λ1+2mn12+2n12+8mm12+2n1.Proof.
From the concept of the first reformulated Zagreb, we have(32)EM1λs⋅λ1,λ2=∑uv∈Eλs⋅λ1,λ2dλs⋅λ1,λ2u+dλs⋅λ1,λ2v−22.
Now, we apply Lemma3,(33)=∑uv∈Eλsdλu+2n2+2+2n1−22+n∑uv∈Eλ1dλ1u+2+dλ1v+2−22+m∑uv∈Eλ2dλ2u+2+dλ2v+2−22+2∑uo∈Vλsv∈Vλ2dλu+2n2+dλ2v+2−22+2∑uN∈Vλsv∈Vλ12+2n1+dλ1v+2−22=∑uv∈Eλsdλu+2n1+n22+n∑uv∈Eλ1dλ1u+dλ1v+22+m∑uv∈Eλ2dλ2u+dλ2v+22+2∑uo∈Vλsv∈Vλ2dλu+dλ2v+2n22+2∑uN∈Vλsv∈Vλ1dλ1v+2+2n12=∑uv∈Eλsdλu2+2n1+n22+2dλu2n1+n2+n∑uv∈Eλ1dλ1u+dλ1v2+4+4dλ1u+dλ1v+m∑uv∈Eλ2dλ2u+dλ2v2+4+4dλ2u+dλ2v+2∑uo∈Vλsv∈Vλ2dλu+dλ2v2+2n22+2dλu+dλ2v2n2+2∑uN∈Vλsv∈Vλ1dλ1v2+2+2n12+2dλ1v2+2n1=∑uv∈Eλsdλu2+4n1+n22∑uv∈Eλs1+4n1+n2∑uv∈Eλsdλu+n∑uv∈Eλ1dλ1u+dλ1v2+4∑uv∈Eλ11+4∑uv∈Eλ1dλ1u+dλ1v+m∑uv∈Eλ2dλ2u+dλ2v2+4∑uv∈Eλ21+4∑uv∈Eλ2dλ2u+dλ2v+2∑uo∈Vλsv∈Vλ2dλu+dλ2v2+4n22∑uo∈Vλsv∈Vλ21+4n2∑uo∈Vλsv∈Vλ2dλu+dλ2v+2∑uN∈Vλsv∈Vλ1dλ1v2+2+2n12∑uN∈Vλsv∈Vλ11+22+2n1∑uN∈Vλsv∈Vλ1dλ1v,=∑uv∈Eλsdλu2+4n1+n22m′+4n1+n2∑uv∈Eλsdλu+nHM1λ1+4nm1+4nM1λ1+mHM1λ2+4mm2+4mM1λ2+2∑uo∈Vλsv∈Vλ2dλu2+dλ2v2+2dλudλ2v+4n22∑uo∈Vλsv∈Vλ21+4n2∑uo∈Vλsv∈Vλ2dλu+∑uo∈Vλsv∈Vλ2dλ2v+2∑uN∈Vλsv∈Vλ1dλ1v2+2+2n12∑uN∈Vλsv∈Vλ11+22+2n1∑uN∈Vλsv∈Vλ1dλ1v=∑uv∈Eλsdλu2+4n1+n22m′+4n1+n2∑uv∈Eλsdλu+nHM1λ1+2∑uo∈Vλsv∈Vλ2dλu2+dλ2v2+2dλudλ2v+4n22∑uo∈Vλsv∈Vλ21+4n2∑uo∈Vλsv∈Vλ2dλu+∑uo∈Vλsv∈Vλ2dλ2v+2∑uN∈Vλsv∈Vλ1dλ1v2+2+2n12∑uN∈Vλsv∈Vλ11+22+2n1∑uN∈Vλsv∈Vλ1dλ1v=∑uv∈Eλsdλu2+4n1+n22m′+4n1+n2∑uv∈Eλsdλu+nHM1λ1+4nm1+4nM1λ1+mHM1λ2+4mm2+4mM1λ2+2∑uo∈Vλsdλu2∑v∈Vλ21+∑uo∈Vλs1∑v∈Vλ2dλv2+2∑uo∈Vλsdλu∑v∈Vλ2dλv+4n22∑uo∈Vλs1∑v∈Vλ21+4n2∑uo∈Vλsdλu∑v∈Vλ21+∑uo∈Vλs1∑v∈Vλ2dλ2v+2∑uN∈Vλs1∑v∈Vλ1dλ1v2+2+2n12∑uN∈Vλs1∑v∈Vλ11+22+2n1∑uN∈Vλs1∑v∈Vλ1dλ1v=∑uv∈Eλsdλu2+4n1+n22m′+4n1+n2∑uv∈Eλsdλu+nHM1λ1+4nm1+4nM1λ1+mHM1λ2+4mm2+4mM1λ2+2n2M1λ+nM1λ2+2⋅2m⋅2m2+4nn2n22+4n22m⋅n2+n⋅2m2+2mM1λ1+2+2n12m⋅n1+22+2n1m⋅2m1=∑uv∈Eλsdλu2+4n1+n2dλu+4n1+n22m′+nHM1λ1+4nm1+4nM1λ1+mHM1λ2+4mm2+4mM1λ2+2n2M1λ+2nM1λ2+16mm2+8nn23+16mn22+16nn2m2+2mM1λ1+2mn12+2n12+8mm12+2n1.
Hence, the proof is done.


## 3. Conclusion

The first and second Zagreb indices, the first multiplicative Zagreb index, the first reformed Zagreb index, and the forgotten topological index were explored in this work, and their exact expressions were investigated. Other degree and distance-based topological indices of these complicated network operations could be calculated in the future work.



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*Source: 6927111-2022-06-07.xml*